diff --git "a/LtFLT4oBgHgl3EQfMS8S/content/tmp_files/load_file.txt" "b/LtFLT4oBgHgl3EQfMS8S/content/tmp_files/load_file.txt" new file mode 100644--- /dev/null +++ "b/LtFLT4oBgHgl3EQfMS8S/content/tmp_files/load_file.txt" @@ -0,0 +1,1326 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf,len=1325 +page_content='A Variant Prescribed Curvature Flow on Closed Surfaces with Negative Euler Characteristic Franziska Borer∗ Peter Elbau† Tobias Weth‡ Abstract On a closed Riemannian surface (M, ¯g) with negative Euler characteristic, we study the problem of finding conformal metrics with prescribed volume A > 0 and the property that their Gauss curvatures fλ = f + λ are given as the sum of a prescribed function f ∈ C∞(M) and an additive constant λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' Our main tool in this study is a new variant of the prescribed Gauss curvature flow, for which we establish local well-posedness and global compactness results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' In contrast to previous work, our approach does not require any sign conditions on f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' Moreover, we exhibit conditions under which the function fλ is sign changing and the standard prescribed Gauss curvature flow is not applicable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' Acknowledgment This work was funded by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation), project 408275461 (Smoothing and Non-Smoothing via Ricci Flow).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' We would like to thank Esther Cabezas–Rivas for helpful discussions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' Introduction Let (M, ¯g) be a two-dimensional, smooth, closed, connected, oriented Riemann manifold endowed with a smooth background metric ¯g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' A classical problem raised by Kazdan and Warner in [11] and [10] is the question which smooth functions f : M → R arise as the Gauss curvature Kg of a conformal metric g(x) = e2u(x)¯g(x) on M and to characterise the set of all such metrics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' For a constant function f, this prescribed Gauss curvature problem is exactly the statement of the Uni- formisation Theorem (see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' [16], [12]): There exists a metric g which is pointwise conformal to ¯g and has constant Gauss curvature Kg ≡ ¯K ∈ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' We now use this statement to assume in the following without loss of generality that the background metric ¯g itself has constant Gauss curvature K¯g ≡ ¯K ∈ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' Furthermore we can normalise the volume of (M, ¯g) to one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' We recall that the Gauss curvature of a conformal metric g(x) = e2u(x)¯g(x) on M is given by the Gauss equation Kg(x) = e−2u(x)(−∆¯gu(x) + ¯K).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='1) Therefore the problem reduces to the question for which functions f there exists a conformal factor u solving the equation − ∆¯gu(x) + ¯K = f(x)e2u(x) in M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='2) Given a solution u, we may integrate (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='2) with respect to the measure µ¯g on M induced by the Riemannian volume form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' Using the Gauss–Bonnet Theorem, we then obtain the identity � M f(x)dµg(x) = � M ¯Kdµ¯g(x) = ¯K vol¯g = ¯K = 2πχ(M), (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='3) where dµg(x) = e2u(x)dµ¯g(x) is the element of area in the metric g(x) = e2u(x)¯g(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' We note that (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='3) immediately yields necessary conditions on f for the solvability of the prescribed Gauss curvature problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' In particular, if ±χ(M) > 0, then ±f must be positive somewhere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' Moreover, if χ(M) = 0, then f must change sign or must be identically zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' ∗Technical University of Berlin, Faculty II—Mathematics and Natural Sciences, Straße des 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' Juni 136, 10623 Berlin, Germany email: borer@tu-berlin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='de †Faculty of Mathematics, University of Vienna, Oskar-Morgenstern-Platz 1, A-1090 Vienna, Austria email: peter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='elbau@univie.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='at ‡Goethe University Frankfurt, Institut f¨ur Mathematik, Robert-Mayer-Straße 10, 60629 Frankfurt, Germany email: weth@math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='uni-frankfurt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='de 1 arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='12015v1 [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='AP] 27 Jan 2023 2 Franziska Borer, Peter Elbau, Tobias Weth In the present paper we focus on the case χ(M) < 0, so M is a surface of genus greater than one and ¯K < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' The complementary cases χ(M) ≥ 0—i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=', the cases where M = S2 or M = T, the 2-torus—will be discussed briefly at the end of this introduction, and we also refer the reader to [18, 19, 2, 8] and the references therein.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' Multiplying equation (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='2) with the factor e−2u and integrating over M with respect to the measure µ¯g, we get the following necessary condition—already mentioned by Kazdan and Warner in [11]—for the average ¯f := 1 vol¯g � M f(x)dµ¯g(x), with vol¯g := � M dµ¯g(x): ¯f = 1 vol¯g � M f(x)dµ¯g(x) = � M (−∆¯gu(x) + ¯K)e−2u(x)dµ¯g(x) = � M (−2|∇¯gu(x)|2 ¯g + ¯K)e−2u(x)dµ¯g(x) < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='4) This condition is not sufficient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' Indeed, it has already been pointed out in [11, Theorem 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='5] that in the case χ(M) < 0 there always exist functions f ∈ C∞(M) with ¯f < 0 and the property that (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='2) has no solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' We recall that solutions of (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='2) can be characterised as critical points of the functional Ef : H1(M, ¯g) → R;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' Ef(u) := 1 2 � M � |∇¯gu(x)|2 ¯g + 2 ¯Ku(x) − f(x)e2u(x)� dµ¯g(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='5) Under the assumption χ(M) < 0, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=', ¯K < 0, the functional Ef is strictly convex and coercive on H1(M, ¯g) if f ≤ 0 and f does not vanish identically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' Hence, as noted in [7], the functional Ef admits a unique critical point uf ∈ H1(M, ¯g) in this case, which is a strict absolute minimiser of Ef and a (weak) solution of (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' The situation is more delicate in the case where fλ = f0 + λ, where f0 ≤ 0 is a smooth, nonconstant function on M with maxx∈M f0(x) = 0, and λ > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' In the case where λ > 0 sufficiently small (depending on f0), it was shown in [7] and [1] that the corresponding functional Efλ admits a local minimiser uλ and a further critical point uλ ̸= uλ of mountain pass type.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' These results motivate our present work, where we suggest a new flow approach to the prescribed Gausss curvature problem in the case χ(M) < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' It is important to note here that there is an intrinsic motivation to formulate the static problem in a flow context.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' Typically, elliptic theories are regarded as the static case of the corresponding parabolic problem;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' in that sense, many times the better-understood elliptic theory has been a source of intuition to generalise the corresponding results in the parabolic case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' Examples of this feedback are minimal surfaces/mean curvature flow, harmonic maps/solutions of the heat equation, and the uniformisation theorem/the two-dimensional normalised Ricci flow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' In this spirit, a flow approach to (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='2), the so-called prescribed Gauss curvature flow, was first introduced by Struwe in [18] (and [2]) for the case M = S2 with the standard background metric and a positive function f ∈ C2(M).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' More precisely, he considers a family of metrics (g(t, ·))t≥0 which fulfils the initial value problem ∂tg(t, x) = 2(α(t)f(x) − Kg(t,·)(x))g(t, x) in (0, T) × M;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='6) g(0, x) = g0(x) on {0} × M, (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='7) with α(t) = � M Kg(t,·)(x)dµg(t,·)(x) � M f(x)dµg(t,·)(x) = 2πχ(M) � M f(x)dµg(t,·)(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='8) This choice of α(t) ensures that the volume of (M, g(t, ·)) remains constant throughout the deformation, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=', � M dµg(t,·)(x) = � M e2u(t,x)dµ¯g(x) ≡ volg0 for all t ≥ 0, where g0 denotes the initial metric on M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' Equivalently one may consider the evolution equation for the associated conformal factor u given by g(t, x) = e2u(t,x)¯g(x): ∂tu(t, x) = α(t)f(x) − Kg(t,·)(x) in (0, T) × M;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='9) u(0, x) = u0(x) on {0} × M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='10) Here the initial value u0 is given by g0(x) = e2u0(x)¯g(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' The flow associated to this parabolic equation is usually called the prescribed Gauss curvature flow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' With the help of this flow, Struwe [18] provided a new proof of a result by Chang and Yang [6] on sufficient criteria for a function f to be the Gauss curvature of a metric g(x) = e2u(x)gS2(x) on S2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' He also proved the sharpness of these criteria.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' In the case of surfaces with genus greater than one, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=', with negative Euler characteristic, the prescribed Gauss curvature flow was used by Ho in [9] to prove that any smooth, strictly negative function on a surface with negative Euler characteristic can be realised as the Gaussian curvature of some metric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' More precisely, Prescribed Curvature Flow on Closed Surfaces with Negative Euler Characteristic 3 assuming that χ(M) < 0 and that f ∈ C∞(M) is a strictly negative function, he proves that equation (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='9) has a solution which is defined for all times and converges to a metric g∞ with Gaussian curvature Kg∞ satisfying Kg∞(x) = α∞f(x) for some constant α∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' While the prescribed Gauss curvature flow is a higly useful tool in the cases where f is of fixed sign, it cannot be used in the case where f is sign-changing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' Indeed, in this case we may have � M f(x)dµg(t,·)(x) = 0 along the flow and then the normalising factor α(t) is not well-defined by (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' As a consequence, a long-time solution of (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='9) might not exist.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' In particular, the static existence results of [7] and [1] can not be recovered and reinterpreted with the standard prescribed Gauss curvature flow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' In this paper we develop a new flow approach to (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='2) in the case χ(M) < 0 for general f ∈ C∞(M), which sheds new light on the results in [7], [1] and [9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' The main idea is to replace the multiplicative normalisation in (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='9) by an additive normalisation, as will be described in details in the next chapter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' At this point, it should be noted that the normalisation factor α(t) in the prescribed Gauss curvature flow given by (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='8) is also not the appropriate choice in the case of the torus, where, as noted before, f has to change sign or be identically zero in order to arise as the Gauss curvature of a conformal metric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' The case of the torus was considered by Struwe in [19], where, in particular, he used to a flow approach to reprove and partially improve a result by Galimberti [8] on the static problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' In this approach, the normalisation in (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='8) is replaced by α(t) = � M f(x)Kg(t,·)(x)dµg(t,·)(x) � M f 2(x)dµg(t,·)(x) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='11) With this choice, Struwe shows that for any smooth u0 ∈ C∗ := � u ∈ H1(M, ¯g) | � M f(x)e2u(x)dµ¯g(x) = 0, � M e2u(x)dµ¯g(x) = 1 � there exists a unique, global smooth solution u of (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='9) satisfying u(t, ·) ∈ C∗ for all t > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' Moreover, u(t, ·) → u∞(·) in H2(M, ¯g) (and smoothly) as t → ∞ suitably, where u∞ + c∞ is a smooth solution of (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='2) for some c∞ ∈ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' In principle, the normalisation (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='11) could also be considered in the case χ(M) < 0, but then the flow is not volume-preserving anymore, which results in a failure of uniform estimates for solutions of (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' Consequently, we were not able to make use of the associated flow in this case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' The paper is organised as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' In Section 2 we set up the framework for the new variant of the prescribed Gauss curvature flow with additive normalisation, and we collect basic properties of it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' In Section 3, we then present our main result on the long-time existence and convergence of the flow (for suitable times tk → ∞) to solutions of the corresponding static problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' In particular, our results show how sign changing functions of the form fλ = f0 + λ arise depending on various assumptions on the shape of f0 and on the fixed volume A of M with respect to the metric g(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' Before proving our results on the time-dependent problem, we first derive, in Section 4, some results on the static problem with volume constraint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' Most of these results will then be used in Section 5, where the parabolic problem is studied in detail and the main results of the paper are proved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' In the appendix, we provide some regularity estimates and a variant of a maximum princple for a class of linear evolution problems with H¨older continuous coefficients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' In the remainder of the paper, we will use the short form f, g(t), u(t), Kg(t), volg(t) := � M dµg(t) = � M e2u(t)dµ¯g, and so on instead of f(x), g(t, x), u(t, x), Kg(t,·)(x), � M dµg(t,·)(x) = � M e2u(t,x)dµ¯g(x), et cetera.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' A New Flow Approach and Some of its Properties Let f ∈ C∞(M) be a smooth function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' We consider now the additive rescaled prescribed Gauss curvature flow given by ∂tu(t) = f − Kg(t) − α(t) = f − e−2u(t)(∆¯gu(t) − ¯K) − α(t) in (0, T) × M, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='1) where α(t) is chosen such that the volume volg(t) of M with respect to g(t) = e2u(t)¯g remains constant along the flow, that is, we require the condition 1 2 d dt volg(t) = � M ∂tu(t)dµg(t) = � M (f − Kg(t) − α(t))dµg(t) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='2) Solving for α(t) then we find α(t) = 1 volg(t) �� M fdµg(t) − ¯K � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' 4 Franziska Borer, Peter Elbau, Tobias Weth So, starting with u0 ∈ Cp,A := � v ∈ W 2,p(M, ¯g) | � M e2vdµ¯g = A � , p > 2, for a given A > 0, we have volg(t) = volg(0) = volg0 = A, for all t ≥ 0, hence we can define αA(t) = 1 A �� M fdµg(t) − ¯K � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='3) Therefore in the following we consider the flow ∂tu(t) = f − Kg(t) − αA(t) in (0, T) × M;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='4) u(0) = u0 ∈ Cp,A on {0} × M, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='5) with αA(t) is chosen like in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' We can now state some first properties of the flow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' Let u be a (sufficiently smooth) solution of (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='4), (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' Then 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' the volume volg(t) of (M, g(t)) is preserved along the flow, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=', volg(t) ≡ volg0 = A for all t ≥ 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' along this trajectory, we have a uniform bound for α given by α(t) ≥ min x∈M f(x) + | ¯K| A =: α1 > −∞ (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='6) and α(t) ≤ max x∈M f(x) + | ¯K| A =: α2 < ∞;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='7) 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' the flow is invariant under adding or subtracting a constant C > 0 to the function f;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' and the energy Ef, defined in (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='5), is decreasing in time along the flow, so Ef(u(t)) ≤ Ef(u0) for all t ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' The first statement directly follows by (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='2) and the choice of α in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' The second one we get since f is smooth and volg(t) = A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' To show the invariance of the flow, let C > 0 be a constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' We then replace f by f ± C in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='4) and see that f ± C − Kg(t) − 1 A �� M (f ± C)dµg(t) − ¯K � = f − Kg(t) − 1 A �� M fdµg(t) − ¯K � = ∂tu(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' So, the flow (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='4) is left unchanged if we replace f by f ± C for a constant C > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' To see that the energy Ef is decreasing along the flow, we use (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='2) and get d dtEf(u(t)) = � M (−��¯gu(t) + ¯K − fe2u(t))∂tu(t)dµ¯g = � M ((−∆¯gu(t) + ¯K)e−2u(t) − f)e2u(t)∂tu(t)dµ¯g = � M ((−∆¯gu(t) + ¯K)e−2u(t) − f)∂tu(t)dµg(t) = � M (Kg(t) − f)∂tu(t)dµg(t) = � M (Kg(t) − f + α(t))∂tu(t)dµg(t) = − � M |∂tu(t)|2dµg(t) ≤ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='8) Therefore on an interval [0, T], we have the uniform a-priori bound Ef(u(T)) + � T 0 � M |∂tu(t)|2dµg(t)dt = Ef(u(0)) (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='9) for any T > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' Prescribed Curvature Flow on Closed Surfaces with Negative Euler Characteristic 5 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' Main Results The following is our first main result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' Let f ∈ C∞(M), p > 2, and u0 ∈ Cp,A for a given A > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' Then the initial value problem (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='4), (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='5) admits a unique global solution u ∈ C([0, ∞);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' C(M)) ∩ C([0, ∞);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' H1(M, ¯g)) ∩ C∞((0, ∞) × M) satisfying the energy bound Ef(u(t)) ≤ Ef(u0) for all t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' Moreover, u is uniformly bounded in the sense that sup t>0 ∥u(t)∥L∞(M,¯g) < ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' Furthermore, as t → ∞ suitably, u converges to a function u∞ in H2(M, ¯g) solving the equation − ∆¯gu + ¯K = fλe2u in M, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='1) where fλ := f + λ with λ = 1 A � ¯K − � M fe2u∞dµ¯g � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='2) In other words, u∞ induces a metric g∞ with Gauss curvature Kg∞ satisfying Kg∞(x) = fλ(x) = f(x) + λ for x ∈ M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='3) Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' For functions f < 0, the convergence of the flow (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='9) is shown in [9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' For the additive rescaled flow (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='4) with initial data (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='5) we get convergence for arbitrary functions f ∈ C∞(M).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' In general we do not have any information about λ and therefore no information about the sign of fλ in Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' On the other hand, more information can be derived for certain functions f ∈ C∞(M) and certain values of A > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' (i) In the case where A ≤ − ¯ K ∥f∥L∞(M,¯g) , it follows that λ = 1 A � ¯K − � M fe2udµ¯g � ≤ ¯K A + ∥f∥L∞(M,¯g) A � M e2udµ¯g = ¯K A + ∥f∥L∞(M,¯g) ≤ 0 for every solution u ∈ C2,A := � v ∈ H2(M, ¯g) | � M e2vdµ¯g = 0 � of the static problem (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='1), and therefore this also applies to λ in Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='1 in this case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' (ii) The following theorems show that fλ in Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='1 may change sign if A > − ¯ K ∥f∥L∞(M,¯g) , so in this case we get a solution of the static problem (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='2) for sign-changing functions f ∈ C∞(M) by using the additive rescaled prescribed Gauss curvature flow (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' Let p > 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' For every A > 0 and c > − ¯ K A there exists ε = ε(c, A, ¯K) > 0 with the following property.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' If u0 ≡ 1 2 log(A) ∈ Cp,A and f ∈ C∞(M) with −c ≤ f ≤ 0 and ∥f + c∥L1(M,¯g) < ε is chosen in Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='1, then the value λ defined in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='2) is positive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' In particular, if f has zeros on M, then fλ in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='3) is sign changing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' Under fairly general assumptions on f, we can prove that λ > 0 if A is sufficiently large and u0 ∈ Cp,A is chosen suitably.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' Let f ∈ C∞(M) be nonconstant with maxx∈M f(x) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' Then there exists κ > 0 with the property that for every A ≥ κ there exists u0 ∈ Cp,A such that the value λ defined in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='2) is positive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' In fact we have even more information on the associated limit u∞ in this case, see Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='8 below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' It remains open how large λ can be depending on A and f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' The only upper bound we have is λ < − � M fdµ¯g, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='4) since we must have ¯fλ = 1 vol¯g � M fλdµ¯g = � M fdµ¯g + λ !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='< 0, so that fλ fulfills the necessary condition (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='4) provided by Kazdan and Warner in [11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' 6 Franziska Borer, Peter Elbau, Tobias Weth 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' The static Minimisation Problem with Volume Constraint To obtain additional information on the limiting function u∞ and the value λ ∈ R associated to it by (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='2) and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='3), we need to consider the associated static setting for the prescribed Gauss curvature problem with the additional condition of prescribed volume.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' Before going into the details of this static problem, we recall an important and highly useful estimate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' The following lemma (see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' [5, Corollary 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='7]) is a consequence of the Trudinger’s inequality [20] which was improved by Moser in [15] (for more details see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' [19, Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='1 and Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='2]): Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' For a two-dimensional, closed Riemannian manifold (M, ¯g) there are constants η > 0 and CMT > 0 such that � M e(u−¯u)dµ¯g ≤ CMT exp � η∥∇¯gu∥2 L2(M,¯g) � (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='1) for all u ∈ H1(M, ¯g) where ¯u := 1 vol¯g � M u dµ¯g = � M u dµ¯g, in view of our assumption that vol¯g = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' As a consequence of Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='1, we have � M epudµ¯g = ep¯u � M e(pu− ¯ pu)dµ¯g ≤ ep¯uCMT exp � η∥∇¯g(pu)∥2 L2(M,¯g) � < ∞ for every u ∈ H1(M, ¯g) and p > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' Consequently, for a given A > 0, the set C1,A := � u ∈ H1(M, ¯g) | V (u) := � M e2udµ¯g = A � (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='2) is well defined and coincides with the closure of C2,A with respect to the H1-norm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' We also note that ¯u ≤ 1 2 log(A) for u ∈ C1,A, (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='3) since by Jensen’s inequality and our assumption that vol¯g = 1 we have 2¯u = − � M 2udµ¯g = � M 2udµ¯g ≤ log � − � e2udµ¯g � = log(A) for u ∈ C1,A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' Furthermore we want to recall the Gagliardo–Nirenberg–Ladyˇzhenskaya interpolation, see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='2 (Gagliardo–Nirenberg–Ladyˇzhenskaya inequality).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' There exists a constant CGNL > 0 such that we have for every ζ ∈ H1(M, ¯g) the inequality ∥ζ∥4 L4(M,¯g) ≤ CGNL∥ζ∥2 L2(M,¯g)∥ζ∥2 H1(M,¯g).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' Now we enter the details of the static prescribed Gauss curvature problem with volume constraint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' In this problem, we wish to find, for given f ∈ C∞(M) and A > 0, critical points of the restriction of the functional Ef defined in (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='5) to the set C1,A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' A critical point u ∈ C1,A of this restriction is a solution of (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='1) for some λ ∈ R, where, here and in the following, we put again fλ := f + λ ∈ C∞(M).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' In other words, such a critical point induces, similarly as the limit u∞ in Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='1, a metric gu with Gauss curvature Kgu satisfying Kgu(x) = fλ(x) = f(x) + λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' The unknown λ ∈ R arises in this context as a Lagrangian multiplier and is a posteriori characterised again by λ = 1 A � ¯K − � M fe2udµ¯g � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' In the study of critical points of the restriction of Ef to C1,A, it is natural to consider the minimisation problem first.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' For this we set mf,A = inf u∈C1,A Ef(u).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' We have the following estimates for mf,A: Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' Let f ∈ C∞(M), A > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' Then we have mf,A ≤ 1 2 � ¯K log(A) − A � M fdµ¯g � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='4) Moreover, if max f ≥ 0, then we have lim sup A→∞ mf,A A ≤ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='5) Prescribed Curvature Flow on Closed Surfaces with Negative Euler Characteristic 7 Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' Let u0(A) ≡ 1 2 log(A), so that � M e2u0(A)dµ¯g = A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' Hence u0(A) is the (unique) constant function in C1,A, and mf,A ≤ Ef(u0(A)) = 1 2 � M (|∇¯gu0(A)|2 ¯g + 2 ¯Ku0(A) − fe2u0(A))dµ¯g = 1 2 � M ( ¯K log(A) − fA)dµ¯g = 1 2 � ¯K log(A) − A � M fdµ¯g � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' This shows (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' To show (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='5), we let ε > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' Since f ∈ C∞(M) and max f ≥ 0 by assumption, there exists an open set Ω ⊂ M with f ≥ −ε on Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' Next, let ψ ∈ C∞(M), ψ ≥ 0, be a function supported in Ω and with ∥ψ∥L∞(M,¯g) = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' Consequently, the set Ω′ := {x ∈ M | ψ > 1} is a nonempty open subset of Ω, and therefore µ¯g(Ω′) > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' Next we consider the continuous function h : [0, ∞) → [0, ∞);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' h(τ) = � M e2τψdµ¯g and we note that h(0) = � M dµ¯g = 1, and that h(τ) ≥ � Ω′ e2τψdµ¯g ≥ e2τµ¯g(Ω′) for τ ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' Hence for every A ≥ 1 there exists 0 ≤ τA ≤ 1 2 � log(A) − log(µ¯g(Ω′)) � (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='6) with h(τA) = A and therefore τAψ ∈ C1,A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' Consequently, mf,A ≤ Ef(τAψ) = 1 2 � M (|∇¯gτAψ|2 ¯g + 2 ¯KτAψ − fe2τAψ)dµ¯g = τ 2 Ac1 − τAc2 − c3 − 1 2 � Ω fe2τAψdµ¯g with c1 = 1 2 � M |∇¯gψ|2 ¯gdµ¯g, c2 = − ¯K � M ψdµ¯g and c3 = 1 2 � M\\Ω fdµ¯g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' Since f ≥ −ε on Ω, we thus deduce that mf,A ≤ τ 2 Ac1 − 2τAc2 + c3 + ε 2 � Ω e2τAψdµ¯g ≤ τ 2 Ac1 − 2τAc2 + c3 + εA 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' Since τA A → 0 as A → ∞ by (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='6), we conclude that lim sup A→∞ mf,A A ≤ ε 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' Since ε > 0 was chosen arbitrarily, (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='5) follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' Let f ∈ C∞(M) nonconstant with maxx∈M f(x) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' For every ε > 0 there exists κ0 > 0 with the following property.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' If A ≥ κ0 and u ∈ C1,A is a solution of − ∆¯gu + ¯K = (f + λ)e2u (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='7) for some λ ∈ R with Ef(u) < εA 2 , then we have λ < ε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' For given ε > 0, we may choose κ0 > 0 sufficiently large so that | ¯ K| 2 log(A) |A| < ε 2 for A ≥ κ0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' Now, let A ≥ κ0, and let u ∈ C1,A be a solution of (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='7) satisfying Ef(u) < εA 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' Integrating (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='7) over M with respect to µ¯g and using that vol¯g(M) = 1 and � M e2udµ¯g = A, we obtain λ = 1 A � ¯K − � M fe2udµ¯g � ≤ − 1 A � M fe2udµ¯g = 1 A � Ef(u) − 1 2 � M (|∇¯gu|2 ¯g + 2 ¯Ku)dµ¯g � ≤ 1 A � Ef(u) + | ¯K|¯u � ≤ ε 2 + | ¯K| 2 log(A) A < ε, as claimed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' Here we used (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='3) to estimate ¯u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' 8 Franziska Borer, Peter Elbau, Tobias Weth Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' Let f ∈ C∞(M) be a nonconstant function with maxx∈M f(x) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' Moreover, let λn → 0+ for n → ∞, and let (un)n∈N be a sequence of solutions of − ∆¯gun + ¯K = (f + λn)e2un in M (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='8) which are weakly stable in the sense that � M (|∇¯gh|2 ¯g − 2(f + λn)e2unh2)dµ¯g ≥ 0 for all h ∈ H1(M).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='9) Then un → u0 in C2(M), where u0 is the unique solution of − ∆¯gu0 + ¯K = fe2u0 in M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='10) Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' We only need to show that (un)n∈N is bounded in C2,α(M) for some α > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='11) Indeed, assuming this for the moment, we may complete the argument as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' Suppose by contradiction that there exists ε > 0 and a subsequence, also denoted by (un)n∈N, with the property that ∥un − u0∥C2(M) ≥ ε for all n ∈ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='12) By (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='11) and the compactness of the embedding C2,α(M) �→ C2(M), we may then pass to a subsequence, still denoted by (un)n∈N, with un → u∗ in C2(M) for some u∗ ∈ C2(M).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' Passing to the limit in (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='8), we then see that u∗ is a solution of (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='10), which by uniqueness implies that u∗ = u0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' This contradicts (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='12), and thus the claim follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' The proof of (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='11) follows by similar arguments as in [7, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' 1063 f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' Since the framework is slightly different, we sketch the main steps here for the convenience of the reader.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' We first note that, by the same argument as in [7, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' 1063 f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='], there exists a constant C0 > 0 with un ≥ −C0 for all n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='13) Since {f < 0} is a nonempty open subset of M by assumption, we may fix a nonempty open subdomain Ω ⊂⊂ {f < 0}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' By [1, Appendix], there exists a constant C1 > 0 with ∥u+ n ∥H1(Ω,¯g) ≤ C1 for all n and therefore � Ω e2undµ¯g ≤ � Ω e2u+ n dµ¯g ≤ C2 for all n (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='14) for some C2 > 0 by the Moser–Trudinger inequality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' Next, we consider a nontrivial, nonpositive function h ∈ C∞ c (Ω) ⊂ C∞(M) and the unique solution w ∈ C∞(M) of the equation −∆¯gw + ¯K = he2w in M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' Moreover, we let wn := un − w, and we note that wn satisfies −∆¯gwn + he2w = (f + λn)e2un in M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' Multiplying this equation by e2wn and integrating by parts, we obtain � M (f + λn)e2(un+wn)dµ¯g = � M � −∆¯gwn + he2w� e2wndµ¯g = � M � 2e2wn|∇¯gwn|2 ¯g + he2(w+wn)� dµ¯g = 2 � M |∇¯gewn|2 ¯gdµ¯g + � Ω he2undµ¯g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='15) Moreover, applying (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='9) to h = ewn gives � M (f + λn)e2(un+wn)dµ¯g ≤ 1 2 � M |∇¯gewn|2 ¯gdµ¯g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='16) Combining (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='14), (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='15) and (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='16) yields ∥∇¯gewn∥2 L2(M,¯g) ≤ −2 3 � Ω he2undµ¯g ≤ 2 3∥h∥L∞(M,¯g)C2 for all n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='17) Prescribed Curvature Flow on Closed Surfaces with Negative Euler Characteristic 9 Next we claim that also ∥ewn∥L2(M,¯g) remains uniformly bounded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' Suppose by contradiction that ∥ewn∥L2(M,¯g) → ∞ as n → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='18) We then set vn := ewn ∥ewn∥L2(M,¯g) , and we note that ∥vn∥L2(M,¯g) = 1 for all n and ∥∇¯gvn∥2 L2(M,¯g) → 0 as n → ∞ (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='19) by (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='17).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' Consequently, we may pass to a subsequence satisfying vn ⇀ v in H1(M, ¯g), where v is a constant function with ∥v∥L2(M,¯g) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='20) However, since ∥ewn∥L2(Ω,¯g) ≤ ∥eun∥L2(Ω,¯g)∥e−w∥L∞(Ω,¯g) ≤ � C2∥e−w∥L∞(Ω,¯g) for all n ∈ N by (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='14) and therefore ∥v∥L2(Ω,¯g) = lim n→∞ ∥vn∥L2(Ω,¯g) = lim n→∞ ∥ewn∥L2(Ω,¯g) ∥ewn∥L2(M,¯g) = 0 by (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='18), we conclude that the constant function v must vanish identically, contradicting (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='20).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' Consequently, ∥ewn∥L2(M,¯g) remains uniformly bounded, which by (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='17) implies that ewn remains bounded in H1(M, ¯g) and therefore in Lp(M, ¯g) for any p < ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' Since eun ≤ ∥ew∥L∞(M,¯g)ewn on M for all n ∈ N, it thus follows that also eun remains bounded in Lp(M, ¯g) for any p < ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' Moreover, by (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='13), the same applies to the sequence un itself.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' Therefore, applying successively elliptic Lp and Schauder estimates to (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='8), we deduce (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='11), as required.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' Let f ∈ C∞(M) be a nonconstant function with maxx∈M f(x) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' Then there exists λ♯ and a C1-curve (−∞, λ♯] → C2(M);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' λ �→ uλ with the following properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' (i) If λ ≤ 0, then uλ is the unique solution of − ∆¯gu + ¯K = fλe2u in M (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='21) and a global minimum of Efλ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' (ii) If λ ∈ (0, λ♯], then uλ is the unique weakly stable solution of (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='21) in the sense of (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='9), and it is a local minimum of Efλ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' (iii) The curve of functions λ �→ uλ is pointwisely strictly increasing on M, and so the volume function (−∞, λ♯] → [0, ∞);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' λ �→ V (λ) := � M e2uλdµ¯g (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='22) is continuous and strictly increasing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' We already know that, for λ ≤ 0, the energy Efλ admits a strict global minimiser uλ which depends smoothly on λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' Moreover, by [1, Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='4], the curve λ �→ uλ can be extended as a C1-curve to an interval (−∞, λ♯] for some λ♯ > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' We also know from [1, Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='4] that, for λ ∈ (−∞, λ♯], the solution uλ is strongly stable in the sense that Cλ := inf h∈H1(M,¯g) 1 ∥h∥2 H1(M,¯g) � M � |∇¯gh|2 ¯g − 2fλe2uλh2� dµ¯g > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='23) Here we note that the function λ �→ Cλ is continuous since uλ depends continuously on λ with respect to the C2-norm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' Next we prove that, after making λ♯ > 0 smaller if necessary, the function uλ is the unique weakly stable solution of (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='21) for λ ∈ (0, λ♯].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' Arguing by contradiction, we assume that there exists a sequence λn → 0+ and corresponding weakly stable solutions (un)n∈N of − ∆¯gun + ¯K = (f + λn)e2un in M (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='24) with the property that un ̸= uλn for every n ∈ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' By Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='5, we know that un → u0 in C2(M).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' Consequently, vn := un − uλn → 0 in C2(M) as n → ∞, whereas the functions vn solve − ∆¯gvn = (f + λn) � e2un − e2uλn � = (f + λn)e2uλn � e2vn − 1 � in M for every n ∈ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='25) 10 Franziska Borer, Peter Elbau, Tobias Weth Combining this fact with (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='23), we deduce that ∥vn∥2 H1(M,¯g) ≤ 1 Cλ � M � |∇¯gvn|2 ¯g − 2(f + λn)e2uλn v2 n � dµ¯g = 1 Cλ � M (f + λn)e2uλn � e2vn − 1 − 2vn � vndµ¯g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' Since vn → 0 in C2(M), there exists a constant C > 0 with |(e2vn − 1 − 2vn)vn| ≤ C|vn|3 on M for all n ∈ N, which then implies with H¨older’s inequality and Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='2 that ∥vn∥2 H1(M,¯g) ≤ C∥(f + λn)e2uλn ∥L∞(M,¯g)∥vn∥3 L3(M,¯g) ≤ C �� M |vn|3· 4 3 dµ¯g � 3 4 = C∥vn∥3 L4(M,¯g) ≤ C∥vn∥3 H1(M,¯g) with a constant C > 0 independent on M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' This contradicts the fact that vn → 0 in H1(M) as n → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' The claim thus follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' It remains to prove that the curve of functions λ �→ uλ is pointwisely strictly increasing on M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' This is a consequence of the uniqueness of weakly stable solutions stated in (ii) and the fact that, as noted in [7], if uλ0 is a solution for some λ0 ∈ (−∞, λ♯], it is possible to construct, via the method of sub- and supersolutions, for every λ < λ0, a weakly stable solution uλ with uλ < uλ0 everywhere in M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' Let f ∈ C∞(M) be nonconstant with maxx∈M f(x) = 0, and let λ♯ > 0 be given as in Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' Then there exists κ1 > 0 with the following property.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' If A ≥ κ1 and u ∈ C1,A is a solution of − ∆¯gu + ¯K = (f + λ)e2u (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='26) for some λ ∈ R with Ef(u) < λ♯A 2 , then 0 < λ < λ♯, and u is not a weakly stable solution of (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='26), so u ̸= uλ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' Let κ0 > 0 be given as in Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='4 for ε = λ♯ > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' Moreover, let κ1 := max � κ0, V (uλ♯) � with V defined in (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='22).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' Next, let u ∈ C1,A be a solution of (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='26) for some λ ∈ R with Ef(u) < λ♯A 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' From Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='4, we then deduce that 0 < λ < λ♯, and by Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='6 (iii) we have u ̸= uλ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' Since uλ is the unique weakly stable solution of (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='26), it follows that u is not weakly stable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' Let p > 2, f ∈ C∞(M) be nonconstant with maxx∈M f(x) = 0, and let λ♯ > 0 be given as in Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' Then there exists κ > 0 with the property that for every A ≥ κ the set ˜C := � u0 ∈ C1,A ∩ W 2,p(M, ¯g) | Ef(u0) < λ♯A 2 � is nonempty, and for every u0 ∈ ˜C the global solution u ∈ C([0, ∞);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' C(M))∩C([0, ∞);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' H1(M, ¯g))∩C∞((0, ∞)× M) of the initial value problem (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='4), (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='5) converges, as t → ∞ suitably, to a solution u∞ of the static problem (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='26) for some λ ∈ (0, λ♯) which is not weakly stable and hence no local minimiser of Efλ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' Let κ1 > 0 be given by Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' By (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='5), there exists κ ≥ κ1 > 0 with mf,A < λ♯A 4 for fixed A > κ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' Consequently, there exists u0 ∈ C1,A ∩ W 2,p(M, ¯g) with Ef(u0) < λ♯A 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' By Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='1, the global solution u ∈ C([0, ∞);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' C(M)) ∩ C([0, ∞);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' H1(M, ¯g)) ∩ C∞((0, ∞) × M) of the initial value problem (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='4), (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='5) converges, as t → ∞ suitably, to a solution u∞ ∈ C1,A of the static problem (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='26) for some λ ∈ R, whereas Ef(u∞) ≤ Ef(u0) < λ♯A 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' Consequently, λ ∈ (0, λ♯) by Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='7, and u∞ is not weakly stable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' Proof of the Main Results 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' Notation and Some Regularity Results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' In this chapter we summarise different kind of estimates which will be useful later.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' In the following, for T > 0 we use the notation Lp t Lr x := Lp([0, T];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' Lr(M, ¯g)) and Lp t Hq x := Lp([0, T];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' Hq(M, ¯g)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' A first regularity result is therefore given by Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' Prescribed Curvature Flow on Closed Surfaces with Negative Euler Characteristic 11 Remark 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' We have ∥θ∥4 Lp t L4x ≤ CGNL∥θ∥2 Lp t L2x∥θ∥2 Lp t H1x for θ ∈ Lp t H1 x with p ∈ [1, ∞].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='2 (Sobolev inequality).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' There exists a constant CS > 0 such that for every ρ ∈ L∞ t H1 x, T ≤ 1, we have ∥ρ∥2 L4 t L4x ≤ CS(∥ρ∥2 L∞ t L2x + ∥∇¯gρ∥2 L2 t L2x) < ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='1) Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' With Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='2 there exists a constant CGNL > 0 such that we have for all T ≤ 1 ∥ρ∥4 L4 t L4x = � T 0 ∥ρ(t)∥4 L4(M,¯g)dt ≤ CGNL � T 0 ∥ρ(t)∥2 L2(M,¯g)∥ρ(t)∥2 H1(M,¯g)dt ≤ CGNL∥ρ∥2 L∞ t L2x � T 0 (∥ρ(t)∥2 L2(M,¯g) + ∥∇¯gρ(t)∥2 L2(M,¯g))dt ≤ CGNL · T ∥ρ∥4 L∞ t L2x + CGNL∥ρ∥2 L∞ t L2x∥∇¯gρ∥2 L2 t L2x ≤ CGNL � ∥ρ∥4 L∞ t L2x + ∥ρ∥2 L∞ t L2x∥∇¯gρ∥2 L2 t L2x � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' By using Young’s inequality we have ∥ρ∥L∞ t L2x∥∇¯gρ∥L2 t L2x ≤ 1 2 � ∥ρ∥2 L∞ t L2x + ∥∇¯gρ∥2 L2 t L2x � and therefore ∥ρ∥2 L4 t L4x ≤ C 1 2 GNL � ∥ρ∥4 L∞ t L2x + 1 4(∥ρ∥2 L∞ t L2x + ∥∇¯gρ∥2 L2 t L2x)2 ≤ C 1 2 GNL(∥ρ∥2 L∞ t L2x + 1 2∥ρ∥2 L∞ t L2x + 1 2∥∇¯gρ∥2 L2 t L2x) ≤ 3 2C 1 2 GNL(∥ρ∥2 L∞ t L2x + ∥∇¯gρ∥2 L2 t L2x) =: CS(∥ρ∥2 L∞ t L2x + ∥∇¯gρ∥2 L2 t L2x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' Since T is finite, ρ ∈ L∞ t H1 x implies that ρ ∈ Lp t H1 x for all p ∈ [1, ∞] which shows that the upper bound is finite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' Furthermore, since T < ∞ and vol¯g = 1, with Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='1 we also have for every p, s ∈ [1, ∞] that Lq tLr x ⊂ Ls tLp x for q ≥ s, r ≥ p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' Since we will often use it in the following, we recall that for v ∈ CtCx := C([0, T], C(M)) we have ∥1 − ev∥2 L∞ t L∞ x ≤ e2∥v∥L∞ t L∞ x ∥v∥2 L∞ t L∞ x (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='2) since for x ∈ R we get with the Taylor expansion |ex − 1| = |1 − ex| ≤ |x|e|x|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='3) Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' With Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='1 we get the following statements: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' For a (sufficiently smooth) solution u of (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='4), (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='5) we have ¯u(t) ≥ 1 2 log � A Cup � =: m0(A, Ef(u0), f, CMT, η1), (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='4) with Cup = CMT exp(4η1(2Ef(u0) + | ¯K| log(A) + A maxx∈M f(x))) where η1 is a number determined by Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' So, especially for a solution u of (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='4), (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='5) we have the uniform bound m0 ≤ ¯u(t) ≤ 1 2 log(A), (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='5) where we used (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='3) and the volume preserving property to get the upper bound of ¯u(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' For a solution u of (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='4), (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='5) we have for all p ∈ R that � M e2pu(t)dµ¯g ≤ Cint(A, CMT, Ef(u0), f, ¯K, η1, η2, p), (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='6) where again, η1, η2 are numbers determined by Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' 12 Franziska Borer, Peter Elbau, Tobias Weth 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' For this part we choose f = f0 where f0 ≤ 0 is a nonconstant, smooth function with maxx∈M f0(x) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' Then there exists a constant Clow = Clow(Cint, f0) > 0 such that � M |f0|dµg(t) ≥ Clow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='7) Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' Let u be a solution of (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='4), (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' We then know that u(t) ∈ CA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' So, with (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='9) we have for all t ≥ 0 that ∥∇¯gu(t)∥2 L2(M,¯g) = 2Ef(u(t)) − � M (2 ¯Ku(t) − fe2u(t))dµ¯g = 2Ef(u(t)) + � M (2| ¯K|u(t) + fe2u(t))dµ¯g ≤ 2Ef(u0) + | ¯K| log(A) + A max x∈M f(x), (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='8) where we used the fact that � M 2u(t)dµ¯g ≤ log(A) by (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='3) and since � M e2u(t)dµ¯g ≡ A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' With this and Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='1 we can now estimate A = � M e2u(t)dµ¯g = e2¯u(t) � M e2(u(t)−¯u(t))dµ¯g ≤ e2¯u(t)CMT exp(η1∥∇¯g(2u(t))∥2 L2(M,¯g)) ≤ e2¯u(t)CMT exp(4η1(2Ef(u0) + | ¯K| log(A) + A max x∈M f(x))) =: Cupe2¯u(t), with Cup = Cup(A, CMT, Ef(u0), f, ¯K, η1) > 0 and therefore ¯u(t) ≥ 1 2 log � A Cup � =: m0(A, CMT, Ef(u0), f, ¯K, η1) ∈ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' So, for a solution u(t) ∈ CA of (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='4), (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='5) we get the uniform bound m0 ≤ ¯u(t) ≤ 1 2 log(A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' Let u be a solution of (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='4), (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' So, u(t) ∈ CA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' With Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='1, (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='5), and (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='8) we directly get for any p ∈ R that � M e2pu(t)dµ¯g = e2p¯u(t) � M e2p(u(t)−¯u(t))dµ¯g ≤ e2p¯u(t)CMT exp(4η2p2∥∇¯gu(t)∥2 L2(M,¯g)) ≤ Cint, (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='9) where Cint = Cint(A, CMT, Ef(u0), f, ¯K, η1, η2, p) > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' Similar to [19, Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='3] we see by the choice of f0, H¨older’s inequality, and (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='9) that 0 < ���� � M � |f0|dµ¯g ���� 2 ≤ � M |f0|e2u(t)dµ¯g � M e−2u(t)dµ¯g ≤ Cint � M |f0|e2u(t)dµ¯g (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='10) which shows the claim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' So, Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='3 is proven.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' Now we can turn to the proofs of the main results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' Short-Time Existence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' Let A > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' We are looking for a short-time solution of (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='4) with initial data (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' Using the Gauss equation (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='1) we can rewrite (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='4), (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='5) in the following way: ∂tu(t) = f − Kg(t) − αA(t) = e−2u(t)∆¯gu(t) + ¯K � 1 A − e−2u(t) � + f − 1 A � M fe2u(t)dµ¯g;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='11) u(0) = u0 ∈ Cp,A := � u ∈ W 2,p(M, ¯g) | � M e2u = A � , (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='12) Prescribed Curvature Flow on Closed Surfaces with Negative Euler Characteristic 13 with p > 2, where αA(t) = 1 A �� M fdµg(t) − ¯K � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' To find a solution of (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='11), (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='12), we consider the linear equation ∂tu(t) = e−2v(t)∆¯gu(t) + ¯K � 1 A − e−2v(t) � + f − 1 A � M fe2v(t)dµ¯g;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='13) u(0) = u0 ∈ Cp,A, (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='14) and use a fixed point argument in the space (X, ∥ · ∥X) := (CtCx, ∥ · ∥CtCx).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' First we observe that for v ∈ CtCx, equation (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='13) is strongly parabolic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' Furthermoren, with p > 2 and the fact that M is compact, we have u0 ∈ Cp,A ⊂ H2(M, ¯g), and therefore u0 ∈ L∞(M, ¯g).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' For the fixed point argument we fix R = R(u0) := ∥u0∥L∞(M,¯g) + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' For fixed T > 0, let X = CtCx = C([0, T], C(M, ¯g)) �→ L∞ t L∞ x with ∥u∥X = max t∈[0,T ], x∈M |u(x, t)|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' For v ∈ X, by [14, Theorem 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='32] and the appendix, we get a unique solution uv ∈ W 2,1 p = W 1,p t Lp x ∩Lp t W 2,p x of (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='13), (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='14) for t ∈ [0, T], x ∈ M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' On XR = {U ∈ X | ∥U∥X ≤ R}, we now define the function Φ as follows: for v ∈ XR, let Φ(v) =: uv be the unique solution of (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='13), (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='14).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' First, we want to show that Φ : XR → XR if T > 0 is chosen small enough.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' If T > 0 is fixed with T ≤ � | ¯K|e2(∥u0∥L∞(M,¯g)+1) + ∥f∥L∞(M,¯g) � 1 + e2(∥u0∥L∞(M,¯g)+1) A ��−1 (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='15) and v ∈ XR, then Φ(v) ∈ XR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' With Proposition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='3 (ii) we directly get ∥Φ(v)∥L∞ t L∞ x = ∥uv∥L∞ t L∞ x ≤ ∥u+ 0 ∥L∞(M,¯g) + TdT (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='16) where dT ≤ | ¯K|e2∥v∥L∞ t L∞ x + ∥f∥L∞(M,¯g) + ∥f∥L∞(M,¯g)e2∥v∥L∞ t L∞ x A ≤ | ¯K|e2R + ∥f∥L∞(M,¯g) � 1 + e2R A � , hence ∥Φ(v)∥L∞ t L∞ x ≤ T � | ¯K|e2R + ∥f∥L∞(M,¯g) � 1 + e2R A �� + ∥u+ 0 ∥L∞(M,¯g) ≤ 1 + ∥u0∥L∞(M,¯g) = R, by (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='15) and since R = ∥u0∥L∞(M,¯g) + 1, which shows the claim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' We now use Schauder’s fixed point Theorem [17] to show the following proposition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' If u0 ∈ Cp,A ⊂ W 2,p(M, ¯g) and T > 0 is fixed with (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='15), then there exists a short-time solution u ∈ X ∩ C∞(M × (0, T)) of (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='11), (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='12).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' Moreover, any such solution satisfies u ∈ C([0, T), H1(M, ¯g)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' Step 1: First we recall Schauder’s Theorem: It asserts that if H is a nonempty, convex, and closed subset of a Banach space B and F is a continuous mapping of H into itself such that F(H) is a relatively compact subset of H, then F has a fixed point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' In our case, B ˆ=X = C([0, T];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' C(M, ¯g)), H ˆ=XR = {u ∈ X | ∥u∥X = ∥u∥CtCx ≤ R}, and F ˆ=Φ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' So to show the existence of a fixed point of Φ in XR, it remains to show that 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' Φ : XR → XR ist continuous and 14 Franziska Borer, Peter Elbau, Tobias Weth 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' Φ(XR) ⊂ XR is relatively compact.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' In a first step we show that Φ : XR → XR ist continuous.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' For this, let (vn)n∈N ⊂ XR be a sequence with ∥vn − v∥X → 0 for n → ∞ with v ∈ XR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' With Proposition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='1 we know that for all vn there exists un ∈ W 2,1 p , p > 2, which is the unique solution of (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='13), (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='14) such that ∥un∥W 2,1 p ≤ C(∥u0∥W 2,p(M,¯g) + ∥dn∥Lp t Lp x) with dn(t) := ¯K � 1 A − e−2vn(t) � + f − 1 A � M fe2vn(t)dµ¯g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' Since vn → v in CtCx and therefore vn → v in L∞ t L∞ x , we know that vn → v in Lp t Lp x for all p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' Furthermore, since the exponential map is continuous, we have e±2vn → e±2v in Lp t Lp x for all p, and therefore dn → d in Lp t Lp x for all p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' Hence, for every ε > 0 there exist NV , Nd ∈ N such that ∥vn − v∥Lp t Lp x < ε for all n ≥ N and ∥dn − d∥Lp t Lp x < ε for all n ≥ N, with N := max{NV , Nd}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' Furthermore we have the estimate ∥e2vn − e2v∥L∞ t L∞ x = ∥(e2vn−2v − 1)e2v∥L∞ t L∞ x ≤ ∥e2vn−2v − 1∥L∞ t L∞ x ∥e2v∥L∞ t L∞ x ≤ ∥2vn − 2v∥e∥2Vn−2V ∥L∞ t L∞ x ∥e2v∥L∞ t L∞ x < 2εe2εe2R, and similarly ∥e−2vn − e−2v∥L∞ t L∞ x < 2εe2εe2R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' Considering now the difference un − u, where un = Φ(vn) and u = Φ(v), we see that un − u fulfils the equation ∂t(un − u)(t) = e−2vn(t)∆¯gun(t) + dn(t) − e−2v(t)∆¯gu(t) − d(t) = e−2vn(t)∆¯g(un − u)(t) + (e−2vn(t) − e−2v(t))∆¯gu(t) + dn(t) − d(t) with ∥un − u∥W 2,1 p ≤ C∥(e−2vn − e−2v)∆¯gu + dn − d∥Lp t Lp x ≤ C � ∥e−2vn − e−2v∥L∞ t L∞ x ∥∆¯gu∥Lp t Lp x + ∥dn − d∥Lp t Lp x � ≤ C(2εe2εe2R∥∆¯gu∥Lp t Lp x + ε) for n ≥ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' Since ∥∆¯gu∥Lp t Lp x is finite and ε > 0 was arbitrary, we see that ∥Φ(vn) − Φ(v)∥W 2,1 p → 0 for n → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' So, we get ∥Φ(vn) − Φ(v)∥X ≤ C∥Φ(vn) − Φ(v)∥Cα ≤ C∥Φ(vn) − Φ(v)∥W 2,1 p → 0 for n → ∞ which shows the continuity of Φ : XR → XR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' In a second step we show that Φ(XR) is relatively compact.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' For this let (un)n∈N ⊂ Φ(XR) be an arbitrary sequence in the image of Φ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' So, again with Proposition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='1, we see that for every un ∈ Φ(XR) there exists a vn ∈ XR with Φ(vn) = un such that ∥un∥W 2,1 p ≤ C(∥u0∥W 2,p(M,¯g) + ∥dn∥Lp t Lp x) ≤ C � ∥u0∥W 2,p(M,¯g) + T| ¯K| A + ∥ ¯Ke−2vn∥Lp t Lp x + ∥f∥Lp t Lp x + ���� 1 A � M fe2vndµ¯g ���� Lp t Lp x � ≤ C � ∥u0∥W 2,p(M,¯g) + T| ¯K| A + | ¯K|e2R + T∥f∥L∞(M,¯g) + T A∥f∥L∞(M,¯g)e2R � ≤ C(A, f, ¯K, R, T, u0) =: Cd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' So, (un)n∈N is uniformly bounded in W 2,1 p ((0, T) × M).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' Using now that W 2,1 p ((0, T) × M) is continuously embedded in Cα([0, T] × M) for some 0 < α < 1 and this on the other hand is compactly embedded in Cβ([0, T] × M) for some 0 < β < α < 1 we can conclude the claim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' We have thus proved that Φ has a fixed point u in XR, which then is a (strong) solution u ∈ W 2,1 p ((0, T) × M) of (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='11), (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='12).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' Step 2: We now show that u ∈ C∞(M × (0, T)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' To see this, we first note the trivial fact that u ∈ W 2,1 p ((0, T)×M) is a strong solution of (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='13), (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='14) with v = u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' Since then v ∈ W 2,1 p ((0, T)×M) ⊂ Cα([0, T]× Prescribed Curvature Flow on Closed Surfaces with Negative Euler Characteristic 15 M), [14, Theorems 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='9 and 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='10] imply the existence of a classical solution ˜u ∈ X ∩ C2+α′,1+α′ loc ((0, T) × M) of (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='13), (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='14) with v = u for some α′ > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' Here C2+α′,1+α′ loc ((0, T) × M) denotes the space of functions f ∈ C2,1((0, T) × M) with the property that ∂tf and all derivatives up to second order of f with respect to x ∈ M are locally α′-H¨older continuous.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' In particular, ˜u ∈ W 2,1 p ((ε, T − ε) × M) for ε ∈ (0, T).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' The function w := u − ˜u ∈ W 2,1 p ((ε, T − ε) × M) is then a strong solution of the initial value problem ∂tw(t) = e−2v(t)∆¯gw(t) for t ∈ (ε, T − ε), w(ε) = u(ε, ·) − ˜u(ε, ·).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' By Proposition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='3 (ii) we then have |w| ≤ ∥u(ε, ·) − ˜u(ε, ·)∥L∞(M,¯g) on (ε, T − ε) × M, whereas ∥u(ε, ·) − ˜u(ε, ·)∥L∞(M,¯g) → 0 as ε → 0 by the continuity of u and ˜u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' It thus follows that u ≡ ˜u on (0, T) × M), and therefore u ∈ C2+α′,1+α′ loc ((0, T) × M).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' Since u solves (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='13), (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='14) with v = u ∈ C2+α′,1+α′ loc ((0, T) × M), we can apply [14, Theorems 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='9] and the above argument again to get u ∈ C4+α′′,2+α′′ loc ((0, T) × M) for some α′′ > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' Repeating this argument inductively, we get u ∈ C k, k 2 loc ((0, T) × M) for every k > 0, and hence u ∈ C∞(M × (0, T)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' Step 3: It remains to show that any solution u ∈ X ∩ C∞((0, T) × M) of (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='11), (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='12) also satisfies u ∈ C([0, T), H1(M, ¯g)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' Since u ∈ C∞((0, T) × M), only the continuity in t = 0 needs to be proved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' Setting φ(t) = ∥u(t)∥2 H1(M,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='¯g) for t ∈ (0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' T),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' we see that 1 2(φ(t2) − φ(t1)) = 1 2 � t2 t1 ∂t∥u(t)∥2 H1(M,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='¯g) dt = � t2 t1 � M � u(t)∂tu(t) + ∇u(t)∇∂tu(t) � dµ¯gdt = � t2 t1 � M � u(t)∂tu(t) − [∆u(t)]∂tu(t) � dµ¯gdt and therefore,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' by H¨older’s inequality,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' 1 2|φ(t2) − φ(t1)| ≤ � t2 t1 � M � |u||∂tu| + |∆u||∂tu| � dµ¯gdt ≤ C∥∂tu∥Lp((0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='T )×M) � ∥u∥Lp((0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='T )×M) + ∥∆u∥Lp((0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='T )×M) � (t2 − t1)β ≤ C∥u∥W 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='2 p ((0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='T )×M)(t2 − t1)β,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' for 0 < t1 < t2 < T with some β > 0 depending on p > 2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' which implies that the function φ is uniformly continuous and therefore bounded on (0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' T).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' We now assume by contradiction that u is not continuous at t = 0 with respect to the H1(M, ¯g)-norm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' Then there exists a sequence (tn)n∈N in (0, T) and ε > 0 with tn → 0+ as n → ∞ and ∥u(tn) − u0∥H1(M,¯g) ≥ ε for all n ∈ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='17) Since ∥u(tn)∥2 H1(M,¯g) = φ(tn) remains bounded as n → ∞, we conclude that, passing to a subsequence, the sequence u(tn) converges weakly in H1(M, ¯g) and therefore strongly in L2(M, ¯g).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' Since the strong L2-limit of u(tn) must be u0 = u(0) as a consequence of the fact that u ∈ X, we deduce that u(tn) ⇀ u0 weakly in H1(M, ¯g) as n → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' Combining this information with Proposition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='1 from the appendix, we deduce that lim sup n→∞ ∥u(tn)∥2 H1(M,¯g) ≤ ∥u0∥2 H1(M,¯g) ≤ lim inf n→∞ ∥u(tn)∥2 H1(M,¯g) (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='18) and therefore ∥u(tn)∥H1(M,¯g) → ∥u0∥H1(M,¯g).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' Note here that this part of Proposition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='1 applies since u solves (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='13), (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='14) with v = u ∈ W 2,1 p ((0, T) × M) ⊂ Cα([0, T] × M) for some α > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' From (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='18) and the uniform convexity of the Hilbert space H1(M, ¯g), we conclude that u(tn) → u0 strongly in H1(M, ¯g), contrary to (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='17).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' Uniqueness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' We now show that the solution from Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='5 is unique.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' Let u0 ∈ W 2,p(M, ¯g), p > 2, and T > 0 be fixed with (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='15).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' Then the short-time solution of u ∈ X ∩ C∞(M × (0, T)) of (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='11), (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='12) given by Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='5 is unique.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' Let u1, u2 ∈ X ∩ C∞(M × (0, T)) be two solutions of (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='11), (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='12).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' The difference u := u1 − u2 ∈ X ∩ C∞(M × (0, T)) then fulfils ∂tu(t) = e−2u1(t)∆¯gu1(t) − e−2u2(t)∆¯gu2(t) − ¯K(e−2u1(t) − e−2u2(t)) − 1 A � M f(e2u1(t) − e2u2(t))dµ¯g = e−2u1(t)∆¯gu(t) + ∆¯gu2(t) � e−2u1(t) − e−2u2(t)� − ¯K(e−2u1(t) − e−2u2(t)) − 1 A � M f(e2u1(t) − e2u2(t))dµ¯g for t ∈ (0, T).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='19) 16 Franziska Borer, Peter Elbau, Tobias Weth In the following, the letter C denotes different positive constants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' Multiplying (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='19) with 2u and integrating over M gives d dt∥u(t)∥2 L2(M,¯g) = 2 � M u(t)∂tu(t)dµ¯g = 2 � M e−2u1(t)u(t)∆¯gu(t)dµ¯g + 2 � M u(t)∆¯gu2(t) � e−2u1(t) − e−2u2(t)� dµ¯g (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='20) − 2 � M ¯Ku(t)(e−2u1(t) − e−2u2(t))dµ¯g − 2 A � M f(e2u1(t) − e2u2(t))dµ¯g � M u(t)dµ¯g ≤ 2 � M e−2u1(t)u(t)∆¯gu(t) + 2 � M V (t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' x)u2(t) + 2ρ(t)∥u(t)∥L2(M,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='¯g) � M |u(t)|dµ¯g ≤ 2 � − � M e−2u1(t)|∇¯gu(t)|2 ¯g + 2 � M e−2u1(t)u(t)⟨∇¯gu1(t),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' ∇¯gu(t)⟩¯gdµ¯g � + 2∥V (t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' ·)∥Lp(M,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='¯g)∥u(t)∥2 L2p′(M,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='¯g) + C∥u(t)∥2 L2(M,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='¯g) ≤ C∥∇¯gu1(t)∥L4(M,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='¯g)∥u(t)∥L4(M,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='¯g)∥���¯gu(t)∥L2(M,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='¯g) + 2∥V (t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' ·)∥Lp(M,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='¯g)∥u(t)∥2 L2p′(M,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='¯g) + C∥u(t)∥2 L2(M,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='¯g) ≤ C � ∥u1(t)∥H2(M,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='¯g)∥u(t)∥2 H1(M,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='¯g) + 2∥V (t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' ·)∥Lp(M,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='¯g)∥u(t)∥2 H1(M,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='¯g) + ∥u(t)∥2 L2(M,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='¯g) � ≤ C � ∥u1(t)∥H2(M,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='¯g) + 2∥V (t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' ·)∥Lp(M,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='¯g) + 1 � ∥u∥2 H1(M,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='¯g),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='21) with functions V ∈ Lp((0, T) × M) ∩ C∞((0, T) × M) and ρ ∈ L∞(0, T).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' Here we used the Sobolev embeddings H1(M, ¯g) �→ Lρ(M) for ρ ∈ [1, ∞).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' Multiplying (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='19) with −2∆u and integrating over M yields d dt∥∇gu(t)∥2 L2(M,¯g) = 2 � M ∇u(t)∇∂tu(t)dµ¯g = −2 � M ∆gu(t)∂tu(t)dµ¯g ≤ −2 � M e−2u1(t)|∆¯gu(t)|2dµ¯g + 2 � M V (x, t)|u(t)||∆u(t)|dµ¯g ≤ −κ∥∆¯gu(t)∥2 L2(M,¯g) + 2∥V (t, ·)∥Lp(M,¯g)∥u∥Lα(M,¯g)∥∆gu∥L2(M,¯g) ≤ −κ∥∆¯gu(t)∥2 L2(M,¯g) + 1 κ∥V (t, ·)∥2 Lp(M,¯g)∥u∥2 Lα(M,¯g) + κ∥∆gu∥2 L2(M,¯g) = 1 κ∥V (t, ·)∥2 Lp(M,¯g)∥u∥2 Lα(M,¯g) ≤ C∥V (t, ·)∥2 Lp(M,¯g)∥u∥2 H1(M,¯g), (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='22) where we used first H¨older’s inequality with α = 2p p−2, then Young’s inequality and finally Sobolev embeddings again.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' Here we note that, by making C > 0 larger if necessary, we may assume that the constants are the same in (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='21) and (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='22).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' Combining these estimates gives d dt∥u(t)∥2 H1(M,¯g) ≤ g(t)∥u(t)∥2 H1(M,¯g) for t ∈ (0, T) (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='23) with the function g ∈ L1(0, T) given by g1(t) = C � ∥u1(t)∥H2(M,¯g) + 3∥V (t, ·)∥Lp(M,¯g) + 1 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' Integrating and using the fact that u ∈ C([0, T), H1(M, ¯g)) by Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='5 with u(0) = u1(0) − u2(0) = 0, we see that ∥u(t)∥2 H1(M,¯g) ≤ � t 0 g(s)∥u(s)∥2 H1(M,¯g) ds for t ∈ [0, T).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' It then follows from Gronwall’s inequality [3] that ∥u(t)∥2 H1(M,¯g) ≡ 0 on [0, T), hence u1 ≡ u2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' Global Existence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' From Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='2 and Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='3 we know that there exists a unique solution u ∈ C([0, T], C(M)) ∩ C([0, 1], H1(M, ¯g)) ∩ C∞((0, T) × M), of the initial value problem (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='11), (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='12).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' In particular we know that u ∈ L∞ t L∞ x for t ∈ [0, T], where T > 0 is given by (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='15).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' In this section we want to show that u posses an L∞-a-priori bound on any time interval [0, T], T < ∞, and therefore, u is the unique global solution of (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='11), (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='12).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' For this we partially follow the idea of [2, Chapter 6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' For every T > 0, there exists M(T) > 0 such that we have sup t∈[0,T ] ∥u(t)∥L∞(M,¯g) ≤ M(T).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' Prescribed Curvature Flow on Closed Surfaces with Negative Euler Characteristic 17 Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' Let I := � t ≥ 0 ��� u is a solution of (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='11) on (0, t] × M with initial data u(0) ∈ Cp,A � , Tmax := sup I, and Tk ⊂ I a sequence in I such that Tk → Tmax for k → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' For any t ∈ [0, Tk] and any xmax(t) ∈ M where u(t, xmax(t)) = max x∈M u(t, x) ≥ 0 we have with ∂tu(t) = ∆g(t)u(t) − e−2u(t) ¯K + f − α(t) and the upper bound for |α| which is given by α0 := max{|α1|, |α2|}, (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='24) that d dt [u(t, xmax(t))] = ∂tu(t, xmax(t)) ≤ | ¯K|e−2u(t,xmax(t)) + f(xmax(t)) + α0 ≤ | ¯K| + ∥f∥L∞(M,¯g) + α0, (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='25) where we used that ∇¯gu(t, xmax(t)) = 0 and therefore d dt [u(t, xmax(t)] = ∂tu(t, xmax(t)) + ∇¯gu(t, xmax(t)) ˙xmax(t) = ∂tu(t, xmax(t)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' Integrating (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='25) on both side with respect to t and taking the supremum over t yields (together with the fact that u(0) = u0 ∈ Cp,A) sup t∈[0,Tk] x∈M u(t, x) ≤ Tk(| ¯K| + ∥f∥L∞(M,¯g) + α0) + sup x∈M u0(x) → Tmax(| ¯K| + ∥f∥L∞(M,¯g) + α0) + sup x∈M u0(x) =: M1(Tmax) < ∞ (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='26) for k → ∞ which shows the upper bound for u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' Analogously, at any point xmin(t) ∈ M where u(t, xmin(t)) = min x∈M u(t, x) ≤ 0 we have with ∂tu(t) = ∆g(t)u(t) − e−2u(t) ¯K + f − α(t), the fact that ¯K < 0, and the upper bound for |α| given by α0 that d dt [u(t, xmin(t))] = ∂tu(t, xmin(t)) ≥ −∥f∥L∞(M,¯g) − α0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='27) Integrating (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='27) on both side with respect to t and taking the infimum over t yields (together with the fact that u(0) = u0 ∈ Cp,A) inf t∈[0,Tk] x∈M u(t, x) ≥ −Tk(∥f∥L∞(M,¯g) + α0) + inf x∈M u0(x) → −Tmax(∥f∥L∞(M,¯g) + α0) + inf x∈M u0(x) =: M2(Tmax) > −∞ (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='28) for k → ∞ which shows the lower bound for u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' So, we get sup t∈[0,T ] x∈M |u(t, x)| ≤ max{|M1(T)|, |M2(T)|} ≤ T(| ¯K| + ∥f∥L∞(M,¯g) + α0) + sup x∈M |u0(x)| =: M(T) (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='29) which shows the claim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' In fact, with the help of (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='9) we can turn (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='29) into a uniform estimate for all time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' Let u be the global, smooth solution of (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='11) with u(0) = u0 ∈ Cp,A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' Then we have that supt>0 ∥u(t)∥L∞(M,¯g) ≤ Cuni < ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' 18 Franziska Borer, Peter Elbau, Tobias Weth Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' We follow the proof of [19, Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' By using the fact that u(t) is a volume preserving solution of (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='11) with u(0) = u0 ∈ Cp,A and therefore � M e2u(t)dµ¯g ≡ A, we get with (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='3) and the fact that ¯K < 0 that Ef(u(t)) = 1 2∥∇¯gu(t)∥2 L2(M,¯g) + � M ¯Ku(t)dµ¯g − 1 2 � M fe2u(t)dµ¯g ≥ ¯K 2 � M 2u(t)dµ¯g − 1 2 � M fe2u(t)dµ¯g ≥ ¯K 2 log(A) − A 2 ∥f∥L∞(M,¯g) > −∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='30) Defining F(t) := � M |∂tu(t)|2dµg(t) = � M |∂tu(t)|2e2u(t)dµ¯g and using the uniform lower bound of Ef given by (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='30), we then get from (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='8) or (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='9), respectively, the estimate � ∞ 0 F(t)dt = � ∞ 0 � M |∂tu(t)|2dµg(t)dt ≤ Ef(u0) + | ¯K| 2 | log(A)| + A 2 ∥f∥L∞(M,¯g).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='31) Hence, for any T > 0 we find tT ∈ [T, T + 1] such that F(tT ) = inf t∈(T,T +1) F(t) ≤ Ef(u0) + | ¯K| 2 | log(A)| + A 2 ∥f∥L∞(M,¯g).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='32) So, at time tT we get with (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='1), H¨olders inequality, (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='6), and (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='32) that ∥∆¯gu(tT )∥L 3 2 (M,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='¯g) ≤ ∥e2u(tT )∂tu(tT )∥L 3 2 (M,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='¯g) + ∥ ¯K∥L 3 2 (M,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='¯g) + ∥e2u(tT )f∥L 3 2 (M,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='¯g) + ∥e2u(tT )α(tT )∥L 3 2 (M,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='¯g) ≤ ∥eu(tT )∥L6(M,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='¯g)F(tT ) 1 2 + | ¯K| + �� M e3u(tT )|f| 3 2 dµ¯g � 2 3 + �� M e3u(tT )|α(tT )| 3 2 dµ¯g � 2 3 ≤ C 1 6 int(A,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' Ef(u0),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' f,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' ¯K,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' η1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' η2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' 3) � Ef(u0) + | ¯K| 2 | log(A)| + A 2 ∥f∥L∞(M,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='¯g) � 1 2 + | ¯K| + C 2 3 int � A,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' Ef(u0),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' f,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' ¯K,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' η1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' η2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' 3 2 � (∥f∥L∞(M,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='¯g) + α0) =: C10 � A,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' Ef(u0),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' f,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' ¯K,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' η1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' η2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' 3 2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' 3 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='33) Furthermore, with Sobolev’s embedding theorem we have W 2, 3 2 ⊂ C0, 2 3 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' Therefore we get with Poincar´e’s inequality, the Calder´on–Zygmund inequality for closed surfaces, and with (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='33) that ∥u(tT ) − ¯u(tT )∥ 3 2 L∞(M,¯g) ≤ C∥u(tT ) − ¯u(tT )∥ 3 2 W 2, 3 2 (M,¯g) ≤ C∥∇2 ¯gu(tT )∥ 3 2 L 3 2 (M,¯g) ≤ C∥∆¯gu(tT )∥ 3 2 L 3 2 (M,¯g) ≤ CC 3 2 10, (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='34) and therefore with (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='5) we obtain the uniform bound ∥u(tT )∥L∞(M,¯g) ≤ CC10 + max � |m0|, 1 2| log(A)| � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='35) Upon shifting time by tT , from (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='29) we now get sup s∈[T +1,T +2] ∥u(s)∥L∞(M,¯g) ≤ sup s∈[tT ,T +2] ∥u(s)∥L∞(M,¯g) ≤ 2(| ¯K| + ∥f∥L∞(M,¯g) + α0) + sup x∈M |u(tT , x)| ≤ 2(| ¯K| + ∥f∥L∞(M,¯g) + α0) + CC10 + max � |m0|, 1 2| log(A)| � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='36) Since T > 0 is arbitrary, the claim follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' Prescribed Curvature Flow on Closed Surfaces with Negative Euler Characteristic 19 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' Convergence of the Flow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' Let f0 ≤ 0 be a smooth, nonconstant function withmaxx∈M f0(x) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' Following here the argumentation of [19], and using (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='31), we know that for a suitable sequence tl → ∞, l → ∞, with associated metrics gl = g(tl) we obtain convergence � M |∂tu(tl)|2dµg(tl) = � M |f0 − Kgl − α(tl)|2dµg(tl) → 0 for l → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='37) Provided that we can also show convergence of the associated sequence of metrics g(tl) to a limit metric g∞ A = e2u∞ A ¯g with Gauss curvature Kg∞ A , it then follows that Kg∞ A = f0 − α∞ A for a constant α∞ A .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' Later we will have a closer look at this constant α∞ A .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' For F(t) = � M |∂tu(t)|2dµg(t) as above, we have F(t) → 0 for t → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' First we consider the evolution equation of the curvature Kg(t) and of α(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' From the Gauss equation (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='1) we get for the curvature that ∂tKg(t) = ∂t(−e−2u(t)∆¯gu(t) + e−2u(t) ¯K) = −2∂tu(t)Kg(t) − ∆g(t)∂tu(t) = 2Kg(t)(Kg(t) − f0 + α(t)) + ∆g(t)(Kg(t) − f0 + α(t)) = 2(Kg(t) − f0 + α(t))2 + 2(f0 − α(t))(Kg(t) − f0 + α(t)) + ∆g(t)(Kg(t) − f0 + α(t)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='38) With (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='3) we get for the evolution equation for α(t): d dtα(t) = 2 A � M f0e2u(t)∂tu(t)dµ¯g = 2 A � M f0(f0 − Kg(t) − α(t))dµg(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='39) So, with (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='38) and (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='39) we arrive at ∂t(Kg(t) − f0 − α(t)) − ∆g(t)(Kg(t) − f0 + α(t)) = 2(Kg(t) − f0 + α(t))2 + 2(f0 − α(t))(Kg(t) − f0 + α(t)) + 2 A � M f0(Kg(t) − f0 + α(t))dµg(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='40) Following the proof of Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='1 in [19] we therefore get 1 2 d dt � M |f0 − Kg(t) − α(t)|2dµg(t) = � M �� ∂tKg(t) + � d dtα(t) �� (Kg(t) − f0 + α(t)) − (Kg(t) − f0 − α(t))3 � dµg(t) = − � M |∇g(t)(Kg(t) − f0 + α(t))|2 g(t)dµg(t) + 2 � M (f0 − α(t))(Kg(t) − f0 + α(t))2dµg(t) + � M (Kg(t) − f0 + α(t))3dµg(t), (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='41) where we used in the second step the fact that � d dtα(t) � � M (Kg(t) − f0 + α(t))dµg(t) = 0 by (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' With H¨older’s inequality we can estimate � M (Kg(t) − f0 + α(t))3dµg(t) ≤ ∥∂tu(t)∥3 L3(M,g(t)) ≤ ∥∂tu(t)∥L2(M,g(t))∥∂tu(t)∥2 L4(M,g(t)) (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='42) 20 Franziska Borer, Peter Elbau, Tobias Weth and by Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='2 we further get with the uniform bound for u ∈ CtCx that ∥∂tu(t)∥2 L4(M,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='g(t)) = �� M |∂tu(t)|4e2u(t)dµ¯g � 1 2 ≤ e∥u∥L∞ t L∞ x ∥∂tu(t)∥2 L4(M,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='¯g) ≤ e∥u∥L∞ t L∞ x � CGNL∥∂tu(t)∥L2(M,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='¯g)∥∂tu(t)∥H1(M,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='¯g) = e∥u∥L∞ t L∞ x � CGNL �� M |∂tu(t)|2e2u(t)e−2u(t)dµ¯g � 1 2 �� M |∂tu(t)|2e2u(t)e−2u(t)dµ¯g + � M |∇¯g∂tu(t)|2 ¯gdµ¯g � 1 2 = e∥u∥L∞ t L∞ x � CGNL �� M |∂tu(t)|2e−2u(t)dµg(t) � 1 2 �� M |∂tu(t)|2e−2u(t)dµg(t) + � M |∇g(t)∂tu(t)|2 g(t)dµg(t) � 1 2 ≤ e∥u∥L∞ t L∞ x max{e∥u∥L∞ t L∞ x ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' e2∥u∥L∞ t L∞ x } � CGNL∥∂tu(t)∥L2(M,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='g(t))∥∂tu(t)∥H1(M,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='g(t)) =: ˜C2∥∂tu(t)∥L2(M,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='g(t))∥∂tu(t)∥H1(M,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='g(t)),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='43) where we used the fact that � M |∇¯g∂tu(t)|2 ¯gdµ¯g = � M |∇g(t)∂tu(t)|2 g(t)dµg(t) =: G(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' Plugging in (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='43) into (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='42) we arrive at � M (Kg(t) − f0 + α(t))3dµg(t) ≤ ˜C2∥∂tu(t)∥2 L2(M,g(t))∥∂tu(t)∥H1(M,g(t)) ≤ ˜C2 2 2 ∥∂tu(t)∥4 L2(M,g(t)) + 1 2∥∂tu(t)∥2 H1(M,g(t)) ≤ ˜C2 2F 2(t) + 1 2(F(t) + G(t)), (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='44) where we used Young’s inequality in the second step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' With α0 = max{|α1|, |α2|} > 0 we furthermore have that 2 � M (f0 − α(t))(Kg(t) − f0 + α(t))2dµg(t) ≤ 2(∥f0∥L∞(M,¯g) + α0)F(t) =: ˜C3(α0, f0)F(t) So, (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='41) yields d dtF(t) + G(t) ≤ 2 � ˜C3F(t) + ˜C2 2F 2(t) + 1 2F(t) � = (2 ˜C3 + 1)F(t) + 2 ˜C2 2F 2(t) =: ˜C4F(t) + 2 ˜C2 2F 2(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='45) We recall that with (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='31) we have lim inft→∞ F(t) = 0 and therefore we know that there exist tl → ∞ with F(tl) → 0 as l → ∞, see (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='37).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' By integrating (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='45) over (tl, t) ⊂ (tl, T) and taking the supremum over (tl, T) we get with � T tl G(t)dt ≥ 0 that sup t∈(tl,T ) F(t) ≤ F(tl) + ˜C4 � T tl F(t)dt + 2 ˜C2 2 � T tl F 2(t)dt ≤ F(tl) + ˜C4 � T tl F(t)dt + 2 ˜C2 2 sup t∈(tl,T ) F(t) � T tl F(t)dt ≤ F(tl) + ˜C4 � T tl F(t)dt + 2 ˜C2 2 sup t∈(tl,T ) F(t) � ∞ tl F(t)dt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' With (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='31) we also have � ∞ tl F(t)dt → 0 for l → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' So, for T > 0 big enough such that for tl < T big enough we have that 2 ˜C2 2 � ∞ tl F(t)dt is small enough to guarantee that 1 − 2 ˜C2 2 � ∞ tl F(t)dt > 0 and therefore the term 2 ˜C2 2 supt∈(tl,T ) F(t) � ∞ tl F(t)dt can be absorbed on the left hand side.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' So, we get sup t∈(tl,T ) F(t) ≤ 1 � 1 − 2 ˜C2 2 � ∞ tl F(t)dt � � F(tl) + ˜C4 � T tl F(t)dt � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' Prescribed Curvature Flow on Closed Surfaces with Negative Euler Characteristic 21 Letting T → ∞ yields sup t∈(tl,∞) F(t) ≤ 1 � 1 − ˜C2 2 � ∞ tl F(t)dt � � F(tl) + ˜C4 � ∞ tl F(t)dt � → 0 as l → ∞ which shows the claim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' To prove now the convergence of the flow, let A > 0 and u0 ∈ Cp,A, p > 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' Furthermore let f ∈ C∞(M) be a smooth, nonconstant function, and (f0, λ) ∈ C∞(M) × R the unique pair such that f = f0 + λ with f0 ≤ 0, f0 nonconstant, and maxx∈M f0(x) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' Since by Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='1 the additive rescaled prescribed Gauss curvature flow (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='4) is invariant under adding or subtracting a constant C > 0 to the function f, for all functions f ∈ {f0 + λ | λ ∈ R} we consider the same flow given by ∂tu(t) = f0 − Kg(t) − αA(t) in (0, T) × M, (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='46) which is (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='4) with f replaced by f0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' With (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='9) we know that 1 2 � M (|∇¯gu(T)|2 ¯g + 2 ¯Ku(T) − f0e2u(T ))dµ¯g = Ef0(u(T)) ≤ Ef0(u(0)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' So, we get with (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='3) that 1 2 � M |∇¯gu(T)|2 ¯gdµ¯g = Ef0(u(T)) − � M ¯Ku(T)dµ¯g + 1 2 � M f0e2u(T )dµ¯g ≤ Ef0(u(T)) + | ¯K| � M u(T)dµ¯g ≤ Ef0(u(0)) + | ¯K| 2 | log(A)|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' So, u is uniformly (in T) bounded in H1(M, ¯g), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=', ∥u∥L∞ t H1x ≤ C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' We now consider ul := u(tl) for a suitable sequence tl → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' By the Theorem of Banach-Alao˘glu we know that (ul)l is weak∗ relatively compact in H1(M, ¯g) and therefore (since H1 is reflexive) also weak relatively compact.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' This means that that there exists a subsequence ulk which we again call ul such that ul → u∞ A weakly in H1(M, ¯g) and therefore strongly in L2(M, ¯g) (by a direct consequence of the Rellich–Kondrachov embedding Theorem).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' Furthermore with (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='6) and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='7) we know that αl := α(tl) → α∞ A as l → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' Moreover we have e±ul → e±u∞ A (as l → ∞) in Lp(M, ¯g) for any 2 ≤ p < ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' Indeed, with Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='8 and (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='3) we have ∥eul − eu∞ A ∥p Lp(M,¯g) = � M epul|1 − eu∞ A −ul|pdµ¯g ≤ epCuni � M |1 − eu∞ A −ul|pdµ¯g ≤ epCuni � M |u∞ A − ul|pep|u∞ A −ul||dµ¯g ≤ epCunie2pCuni � M |u∞ A − ul|p−2|u∞ A − ul|2dµ¯g ≤ e3pCuni(2Cuni)p−2∥u∞ A − ul∥2 L2(M,¯g) → 0 as l → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' Replacing ul by −ul we get also e−ul → e−u∞ A in Lp(M, ¯g) as l → ∞ for any p < ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' Moreover, with Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='8 and Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='9 we also have e2ul∂tul → 0 in L2(M, ¯g) as l → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' Furthermore we have ∥e2ulαl − e2u∞ A α∞ A ∥L2(M,¯g) ≤ ∥e2ul(αl − α∞ A )∥L2(M,¯g) + ∥α∞ A (e2ul − e2u∞ A )∥L2(M,¯g) ≤ ∥e2ul∥L∞(M,¯g)|αl − α∞ A |A 1 2 + |α∞ A |∥e2ul − e2u∞ A ∥L2(M,¯g) → 0 for l → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' So, considering our evolution equation (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='11), we therefore get ∆¯gul = e2ul∂tul + ¯K − e2ulf0 + e2ulαl → ¯K − e2u∞ A f0 + e2u∞ A α∞ A =: (∆¯gu)∞ A 22 Franziska Borer, Peter Elbau, Tobias Weth in L2(M, ¯g).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' Since the Laplace operator ∆¯g is closed we know that (∆¯gu)∞ A = ∆¯gu∞ A .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' Hence ∥∆¯g(ul − u∞ A )∥L2(M,¯g) → 0 as l → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' So, we even have strong convergence ul → u∞ A in H2(M, ¯g) and uniformly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' Thus, passing to the limit l → ∞ in the equation e2ul∂tul − ∆¯gul = − ¯K + e2ulf0 − e2ulαl we get the identity −∆¯gu∞ A = − ¯K + e2u∞ A f0 − e2u∞ A α∞ A and therefore Kg∞ A = f0 − α∞ A = f0 + 1 A � ¯K + � M |f0|dµg∞ A � which shows the convergence of the flow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' The Sign of the Constant α∞ A .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' In this subsection we prove Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='3 and Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='4, with other words, under certain assumptions we can now further estimate the expression 1 A � ¯K + � M |f0|dµg∞ A � to show that it is positive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' The proof of Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='4 is already covered by the proof of Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' So we can turn to Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' Proof of Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' We have seen in Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='7 that in the case where u0 ≡ 1 2 log(A) ∈ Cp,A, the uniform L∞-bound on the global solution of the initial value problem (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='11), (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='12) only depends on A and an upper bound on ∥f∥L∞(M,¯g).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' In other words, if A > 0 and c > 0 are fixed, then there exists τ > 0 with the property that sup t>0 ∥u(t)∥L∞(M,¯g) ≤ τ for every f ∈ C∞(M) with ∥f∥L∞(M,¯g) ≤ c and the corresponding solution u of the initial value problem (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='11), (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='12) with u0 ≡ 1 2 log(A) ∈ Cp,A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' Consequently, we also have ∥u∞∥L∞(M,¯g) ≤ τ under the current assumptions on f, which implies that λ = 1 A � ¯K − � M fe2u∞dµ¯g � = 1 A � ¯K + cA − � M (f + c)e2u∞dµ¯g � ≥ c + ¯K A − ∥f + c∥L1(M,¯g)∥e2u∞∥L∞(M,¯g) ≥ c + ¯K A − ∥f + c∥L1(M,¯g)e2τ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' Hence, if ∥f + c∥L1(M,¯g) < ε := c+ ¯ K A e2τ , we have λ > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' Appendix As before, let (M, ¯g) be a two-dimensional, smooth, closed, connected, oriented Riemann manifold endowed with a smooth background metric ¯g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' For a domain Ω ⊂ M × R and p ≥ 1, we let W 2,1 p (Ω) denote the space of functions u ∈ Lp(Ω) which have weak derivatives Du, D2u and ∂tu in Lp(Ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' In the following, we fix p > 2, which implies that W 2,1 p (Ω) is continuously embedded in Cα(Ω) for some α = α(p) > 0, (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='1) see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' [13, Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' We consider the linear parabolic problem ∂tu(x, t) = a(x, t)∆¯gu(x, t) + c(x, t)u(x, t) + d(x, t), (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='2) with a, c, d ∈ C(Ω) and d ∈ Lp(Ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' We say that a function u ∈ W 2,1 p (Ω) is a (strong) solution of (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='2) in Ω if (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='2) holds almost everywhere in Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' Specifically, we consider (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='2) on the cylindrical domains ΩT = M × (0, T) and �ΩT = M × (−∞, T) in the following.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' In particular, we consider strong solutions of (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='2) together with the initial condition u(0, x) = u0(x) in M (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='3) with u0 ∈ W 2,p(M, ¯g), which is supposed to hold in the (initial) trace sense.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' Prescribed Curvature Flow on Closed Surfaces with Negative Euler Characteristic 23 Proposition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' Let T > 0, a, c ∈ C(ΩT ) with aT := min (x,t)∈ΩT a(x, t) > 0, let d ∈ Lp(ΩT ) for some p > 2, and let u0 ∈ W 2,p(M, ¯g).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' Then the initial value problem (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='2), (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='3) has a unique strong solution u ∈ W 2,1 p (ΩT ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' Moreover, u satisfies the estimate ∥u∥W 2,1 p (ΩT ) ≤ C � ∥u0∥W 2,p(M,¯g) + ∥d∥Lp(ΩT ) � (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='4) with a constant C > 0 depending only on ∥a∥L∞(ΩT ), ∥c∥L∞(ΩT ) and aT .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' Moreover, C does not increase after making T smaller.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' If, moreover, a, c, d ∈ Cα(ΩT ) for some α > 0, then u ∈ C(ΩT )∩C2,1(ΩT ) is a classical solution of (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='2), (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='3), and we have the inequality ∥u0∥H1(M,¯g) ≥ lim sup t→0+ ∥u(t)∥H1(M,¯g) (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='5) Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' In the following, the letter C stands for various positive constants depending only on ∥a∥L∞(ΩT ), ∥c∥L∞(ΩT ), and aT , and which do not increase after making T smaller.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' Step 1: We first assume that we are given a strong solution u ∈ W 2,1 p (ΩT ) of (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='2), (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='3) with u0 ≡ 0 ∈ W 2,p(M, ¯g).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' We then define v : �ΩT → R by v(x, t) = � u(x, t), for t > 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' 0, for t ≤ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' Then v ∈ W 2,1 p (�ΩT ) solves (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='2) with a, c, d replaced by suitable extensions ˜a, ˜c, ∈ L∞(�ΩT ), ˜d ∈ Lp(�ΩT ) satisfying ˜a(x, t) = a(x, 0), ˜c(x, t) = c(x, 0) and ˜d(x, t) = 0 for t ≤ 0, x ∈ M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' Therefore, [14, Theorem 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='22] gives rise to the uniform bound ∥D2v∥Lp(�ΩT ) + ∥∂tv∥Lp(�ΩT ) ≤ C � ∥ ˜d∥Lp(�ΩT ) + ∥v∥Lp(�ΩT ) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='6) This translates into the estimate ∥D2u∥Lp(ΩT ) + ∥∂tu∥Lp(ΩT ) ≤ C � ∥d∥Lp(ΩT ) + ∥u∥Lp(ΩT ) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='7) Moreover, setting V (t) := ∥u(t)∥p Lp(M,¯g) for t ∈ R, we have V (0) = 0 and ˙V (t) = p � M |u(t)|p−2u(t)∂tu(t)dµ¯g ≤ pV (t) 1 p′ ∥∂tu(t)∥Lp(M,¯g) ≤ p � V (t) p′ + ∥∂tu(t)∥p Lp(M,¯g) p � = p p′ V (t) + ∥∂tu(t)∥p Lp(M,¯g) for t ∈ (0, T), therefore V (t) = � t 0 ˙V (s) ds ≤ p p′ � t 0 V (s) ds + ∥∂tu∥p Lp(Ωt) ≤ p p′ � t 0 V (s) ds + C � ∥d∥p Lp(Ωt) + ∥u∥p Lp(Ωt) � ≤ C �� t 0 V (s) ds + ∥d∥p Lp(Ωt) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' By Gronwall’s inequality we get V (t) ≤ C∥d∥p Lp(Ωt) and thus ∥u(t)∥Lp(M,¯g) ≤ C∥d∥Lp(Ωt) for t ∈ [0, T].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='8) This already implies the uniqueness of strong solutions of (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='2), (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='3), since the difference u of two solutions u1, u2 ∈ W 2,1 p (ΩT ) of (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='2), (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='3) satisfies (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='2), (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='3) with u0 = 0 and d = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' Moreover, if u ∈ W 2,1 p (ΩT ) is a strong solution of (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='2), (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='3), then the function ˆu ∈ W 2,1 p (ΩT ) given by ˆu(x, t) := u(x, t) − u0(x) safisfies (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='2), (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='3) with u0 = 0 and d replaced by ˆd given by ˆd(x, t) = d(x, t) + a(x, t)∆¯gu0(x) + c(x, t)u0(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' Consequently, combining (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='7) and (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='8), and using an interpolation estimate for Du, we find that ∥u∥W 2,1 p (ΩT ) ≤ ∥ˆu∥W 2,1 p (ΩT ) + ∥u0∥W 2,p(M,¯g) ≤ C � ∥ ˆd∥Lp(ΩT ) + ∥ˆu∥Lp(ΩT ) � + ∥u0∥W 2,p(M,¯g) ≤ C∥ ˆd∥Lp(ΩT ) + ∥u0∥W 2,p(M,¯g) ≤ C � ∥d∥Lp(ΩT ) + ∥u0∥W 2,p(M,¯g) � , 24 Franziska Borer, Peter Elbau, Tobias Weth as claimed in (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' Step 2 (Existence): In the case where a, c, d ∈ Cα(ΩT ) and u0 ∈ C2+α(M), the existence of a classical solution u ∈ C(ΩT ) ∩ C2,1(ΩT ) of (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='2), (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='3) follows as in [14, Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' In the general case we consider (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='2), (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='3) with coefficients an, cn, dn ∈ Cα(ΩT ), u0,n ∈ C2+α(M), in place of a, c, d, u0 with the property that an → a, cn → c in L∞(ΩT ), dn → d ∈ Lp(ΩT ) as well as u0,n → u0 in W 2,p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' The associated unique solutions un ∈ C(ΩT ) ∩ C2,1(ΩT ) are uniformly bounded in W 2,1 p (ΩT ) by (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='4), and therefore we have un ⇀ u in W 2,1 p (ΩT ) after passing to a subsequence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' For every φ ∈ C∞ c (ΩT ), we then have � ΩT � ∂tu(x, t) − a(x, t)∆¯gu(t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='x) − c(x, t)u(x, t) − d(x, t) � φ(x, t)dµ¯g(x)dt = lim n→∞ � ΩT � ∂tun(x, t) − an(x, t)∆¯gun(x, t) − cn(x, t)un(x, t) − dn(x, t) � φ(x, t)dµ¯g(x)dt = 0, and from this we deduce that ∂tu(x, t) − a(x, t)∆¯gu(x, t) − c(x, t)u(x, t) − d(x, t) = 0 almost everywhere in ΩT , so u is a strong solution of (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' Step 3: It remains to show the inequality (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='5) in the case where a, c, d ∈ Cα(ΩT ) for some α > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' Since u ∈ C(ΩT ) ∩ C2,1(ΩT ) in this case and therefore ∥u0∥L2(M,¯g) = lim t→0+ ∥u(t)∥L2(M,¯g), it suffices to show that ∥∇u0∥L2(M,¯g) ≥ lim sup t→0+ ∥���u(t)∥L2(M,¯g).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='9) If u0 ∈ C2+α(M) for some α > 0, this follows by [14, Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='14] with lim in place of lim sup, since the function t �→ u(t) is continuous from [0, T) → C2+α(M) in this case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' Moreover,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' in this case we have,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' by H¨older’s and Young’s inequality,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' d dt∥∇u(t)∥2 L2(M,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='¯g) = − � M ∂tu(t)∆u(t)dµ¯g = − � M � a(t)|∆u(t)|2 + c(t)u(t)∆u(t) + d(t)∆u(t) � dµ¯g ≤ −aT ∥∆¯gu(t)∥2 L2(M,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='¯g) + ∥c(t)u(t) + d(t)∥L2(M,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='¯g)∥∆¯gu(t)∥L2(M,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='¯g) ≤ −aT ∥∆¯gu(t)∥2 L2(M,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='¯g) + aT ∥∆¯gu(t)∥2 L2(M,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='¯g) + 1 4aT ∥c(t)u(t) + d(t)∥2 L2(M,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='¯g) = 1 4aT ∥c(t)u(t) + d(t)∥2 L2(M,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='¯g),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' and therefore ∥∇u(t)∥2 L2(M,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='¯g) ≤ ∥∇u(0)∥2 L2(M,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='¯g) + 1 4aT � t 0 ∥c(s)u(s) + d(s)∥2 L2(M,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='¯g) ds for t > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='10) In the general case, we consider (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='2), (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='3) with a sequence of initial conditions un,0 in place of u0, where un,0 → u0 in H2(M).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' The associated unique solutions un ∈ C(ΩT ) ∩ C2,1(ΩT ) are uniformly bounded in W 2,1 p (ΩT ) by (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='4), and they are also uniformly bounded in C2,1([ε, T] × M) by [14, Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='15] for every ε ∈ (0, T).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' Fix t ∈ (0, T).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' Passing to a subsequence, we may assume that un ⇀ u in W 2,1 p (ΩT ), un → u strongly in C0(ΩT ) and un(t) → u(t) strongly in C1(M).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' As in Step 2, we see, by testing with φ ∈ C∞ c (ΩT ), that ∂tu(x, t) − a(x, t)∆¯gu(x, t) − c(x, t)u(x, t) − d(x, t) = 0 almost everywhere in ΩT , so u is the unique strong solution of (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='2), (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' Moreover, by (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='10) we have ∥∇u(t)∥2 L2(M,¯g) = lim n→∞ ∥∇un(t)∥2 L2(M,¯g) ≤ lim n→∞ � ∥∇un(0)∥2 L2(M) + 1 4aT � t 0 ∥c(s)un(s) + d(s)∥2 L2(M,¯g) ds � = ∥∇u(0)∥2 L2(M,¯g) + 1 4aT � t 0 ∥c(s)u(s) + d(s)∥2 L2(M,¯g) ds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' It thus follows that ∥∇u(t)∥2 L2(M,¯g) − ∥∇u(0)∥2 L2(M,¯g) ≤ 1 4aT � t 0 ∥c(s)u(s) + d(s)∥2 L2(M,¯g) ds Prescribed Curvature Flow on Closed Surfaces with Negative Euler Characteristic 25 and therefore lim sup t→0 � ∥∇u(t)∥2 L2(M,¯g) − ∥∇u(0)∥2 L2(M,¯g) � ≤ 1 4aT lim t→0+ � t 0 ∥c(s)u(s) + d(s)∥2 L2(M,¯g) ds = 0, as claimed in (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' Next we prove a maximum principle for solutions of (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='2), (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' We need the following preliminary lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' Let T > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' (i) For any function u ∈ C2(M) we have � {x∈M|u(x)>0} ∆¯gudµ¯g ≤ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' (ii) Let u, ρ ∈ C1([0, T]) be functions with u(0) ≤ 0 and ρ(T) ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' Then � {t∈[0,T ]|u(t)>0} � ρ(t)∂tu(t) + κu(t) � dt ≥ 0 with κ := sup s∈(0,T ) ∂tρ(s).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='11) (iii) Let u ∈ C2,1(ΩT ) ∩ C0,1(ΩT ), ρ ∈ C0,1(ΩT ) be functions with u ≤ 0 on M × {0} and ρ ≥ 0 on M × {T}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' Then we have � {(x,t)∈M×[0,T ]|u(x,t)>0} (ρ(x, t)∂tu(x, t) + κu(x, t) − ∆¯gu(x, t))dµ¯g(x)dt ≥ 0 with κ := sup (s,x)∈M×(0,T ) ∂tρ(s, x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='12) Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' (i) By Lebesgue’s theorem, it suffices to prove � {x∈M|u(x)>εn} ∆¯gudµ¯g ≤ 0 (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='13) for a sequence εn → 0+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' By Sard’s Lemma, we may choose this sequence such that Ωε := {x ∈ M | u(x) > εn} is an open set of class C1, whereas the outer unit vector field of Ωε is given by (x, t) �→ − ∇¯gu(x,t) |∇¯gu(x,t)|¯g .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' Hence (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='13) follows from the divergence theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' (ii) The set {t ∈ [0, T] | u(t) > 0} is a union of at most countably many open intervals Ij, j ∈ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' For any such interval, partial integration gives � Ij � ρ(t)∂tu(t) + ∂tρ(t)u(t) � dt = � 0, if T ̸∈ Ij;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' ρ(T)u(T) ≥ 0, if T ∈ Ij.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' Consequently, � {t∈[0,T ]|u(t)>0} ρ(t)∂tu(t) dt ≥ − � {t∈[0,T ]|u(t)>0} ∂tρ(t)u(t) dt ≥ − � {t∈[0,T ]|u(t)>0} κu(t) dt with κ given in (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='11).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' This shows the claim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' (iii) This is a direct consequence of (i), (ii) and Fubini’s theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' Proposition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' (Maximum principle) Let T > 0, a, c ∈ C(ΩT ) with aT := min (x,t)∈ΩT a(x, t) > 0, let d ∈ Lp(ΩT ) for some p > 2 with dT := sup(x,t)∈ΩT d(x, t) < ∞, and let u0 ∈ W 2,p(M, ¯g).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' Moreover, let u ∈ W 2,1 p (ΩT ) be the unique solution of (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='2), (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' (i) If u0 ≤ 0 on M and dT ≤ 0, then u ≤ 0 on ΩT .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' (ii) If c ≡ 0 on ΩT , then u(x, t) ≤ ∥u+ 0 ∥L∞(M,¯g) + tdT for t ∈ [0, T], x ∈ M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='14) 26 Franziska Borer, Peter Elbau, Tobias Weth Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' (i) Step 1: We consider the special case a ∈ C0,1(ΩT ), u0 ≤ 0 and dT ≤ −ε for some ε > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' We put ρ := 1 a ∈ C0,1(ΩT ) and κ := sup (s,x)∈M×(0,T ) ∂tρ(s, x) as in (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='12).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' Moreover, we consider the function ˘u ∈ W 2,1 p (ΩT ), ˘u(x, t) = e−˘κtu(x, t) with ˘κ = |κ| min(x,t)∈ΩT ρ(x,t) + ∥c∥L∞(ΩT ), noting that ˘u satisfies ρ(x, t)∂t˘u(x, t) − ∆¯g˘u(x, t) + κ˘u(x, t) = e−˘κt� u(x, t)(ρ(x, t)c(x, t) − ρ(x, t)˘κ + κ) + ρ(x, t)d(x, t) � ≤ −ρ(x, t)εe−˘κt almost everywhere in {(x, t) ∈ ΩT | ˘u(x, t) > 0}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='15) We now let (un)n∈N be a sequence in C2,1(ΩT ) ∩ C0,1(ΩT ) with un(x, 0) ≤ 0 and un → ˘u in W 2,1 p (ΩT ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' Since the functions gn := 1{(x,t)∈M×[0,T ]|un(x,t)>0} are bounded in Lp′(ΩT ), we may pass to a subsequence such that gn ⇀ g in Lp′(ΩT ), where g ≥ 0 and g ≡ 1 in {(x, t) ∈ M × [0, T] | ˘u(x, t) > 0}, since un → ˘u uniformly as a consequence of (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='1) and therefore gn → 1 pointwisely on {(x, t) ∈ M × [0, T] | ˘u(x, t) > 0}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' Applying Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='2 (iii) to un,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' we find that 0 ≤ � {(x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='t)∈M×[0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='T ]|un(x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='t)>0} � ρ(x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' t)∂tun(t) − ∆¯gun(x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' t) + κun(x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' t) � dµ¯g(x)dt = � M×(0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='T ) gn(x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' t) � ρ(x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' t)∂tun(x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' t) − ∆¯gun(x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' t) + κun(x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' t) � dµ¯g(x)dt for all n ∈ N and therefore 0 ≤ lim n→∞ � M×(0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='T ) gn(x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' t) � ρ(x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' t)∂tun(x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' t) − ∆¯gun(x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' t) + κun(x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' t) � dµ¯g(x)dt = � M×(0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='T ) g(x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' t) � ρ(x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' t)∂t˘u(x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' t) − ∆¯g˘u(x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' t) + κ˘u(x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' t) � dµ¯gdt ≤ − � M×(0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='T ) g(x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' t)ρ(x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' t)εe−˘κtdµ¯g(x)dt ≤ − � {(x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='t)∈M×(0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='T )|˘u(x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='t)>0} ρ(x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' t)εe−˘κtdµ¯g(x)dt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' We thus conclude that {(x, t) ∈ M × (0, T) | ˘u(x, t) > 0} = {(x, t) ∈ M × (0, T) | u(x, t) > 0} = ∅ and therefore u ≤ 0 in M × (0, T).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' Step 2: In the special case where a ∈ C0,1(ΩT ), u0 ≤ 0 and dT ≤ 0, we may apply Step 1 to the functions uε ∈ W 2,1 p (ΩT ) defined by uε(x, t) = u(x, t) − εt, which yields that uε ≤ 0 for every ε > 0 and therefore u ≤ 0 in ΩT .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' Step 3: In the general case, we consider a sequence an ∈ C0,1(ΩT ) with an → a in C(ΩT ), and we let un denote the associated solutions of (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='2), (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='3) with a replaced by an.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' As in the end of the proof of Proposition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='1, we then find that, after passing to a subsequence, un ⇀ ˜u in W 2,1 p (ΩT ), where ˜u is a solution of (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='2), (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' By uniqueness, we have u = ˜u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' Moreover, since un ≤ 0 for all n by Step 3, we have u = ˜u ≤ 0, as required.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' (ii) We consider the function v ∈ W 2,1 p (ΩT ) given by v(x, t) = u(x, t)−∥u+ 0 ∥L∞(M) −tdT , which, by assumption, satisfies (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='2), (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='3) with c ≡ 0, d − dT in place of d and u0 − ∥u+ 0 ∥L∞(M) in place of u0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' Then (i) yields v ≤ 0 in ΩT , and therefore u satisfies (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='14).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' References [1] F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' Borer, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' Galimberti, and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' Struwe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' “Large” conformal Metrics of prescribed Gauss Curvature on Surfaces of higher Genus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' Commentarii Mathematici Helvetici 90.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='2 (2015), pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' 407–428.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='4171 /CMH/358.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' [2] R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' Buzzano, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' Schulz, and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' Struwe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' Variational Methods in Geometric Analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' (to appear).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' [3] T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' Cazenave, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' Haraux, and Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' Martel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' An Introduction to Semilinear Evolution Equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' Oxford Lecture Series in Mathematics and its Application 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' The Clarendon Press, Oxford University Press, 1999.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' [4] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' Ceccon and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' Montenegro.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' Optimal Lp-Riemannian Gagliardo-Nirenberg inequalities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' Mathematische Zeitschrift 258.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='4 (2008), pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' 851–873.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' issn: 0025-5874.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='1007/s00209-007-0202-8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' [5] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='-Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' Chang.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' Non-linear elliptic equations in conformal geometry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' Zurich Lectures in Advanced Math- ematics 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' European Mathematical Society, 2004.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' Prescribed Curvature Flow on Closed Surfaces with Negative Euler Characteristic 27 [6] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='-Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' Chang and P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' Yang.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' Prescribing Gaussian curvature on S2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' Acta Mathematica 159.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='1 (1987), pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' 215–259.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' issn: 1871-2509.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='1007/BF02392560.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' [7] W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='-Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' Ding and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='-Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' Liu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' A Note on the Problem of Prescribing Gaussian Curvature on Surfaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' Trans- actions of the American Mathematical Society 347.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='3 (1995), pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' 1059–1066.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' issn: 00029947.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' url: http: //www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='jstor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='org/stable/2154889.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' [8] L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' Galimberti.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' Compactness issues and bubbling phenomena for the prescribed Gaussian curvature equation on the torus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' Calculus of Variations and Partial Differential Equations 54.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='3 (2015), pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' 2483–2501.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='1007/s00526-015-0872-8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' [9] P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' Ho.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' Prescribed Curvature Flow on Surfaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' Indiana University Mathematics Journal 60.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='5 (2011), pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' 1517–1541.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' issn: 00222518, 19435258.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' url: http://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='jstor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='org/stable/24903835.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' [10] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' Kazdan and F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' Warner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' Curvature Functions for Open 2-Manifolds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' Annals of Mathematics 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='2 (1974), pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' 203–219.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' issn: 0003486X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' url: http://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='jstor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='org/stable/1970898.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' [11] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' Kazdan and F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' Warner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' Curvature Functions for Compact 2-Manifolds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' Annals of Mathematics 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='1 (1974), pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' 14–47.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' issn: 0003486X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' url: http://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='jstor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='org/stable/1971012.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' [12] P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' Koebe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' ¨Uber die Uniformisierung beliebiger analytischer Kurven (Dritte Mitteilung).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' Nachrichten von der Gesellschaft der Wissenschaften zu G¨ottingen, Mathematisch-Physikalische Klasse 1 (1908), pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' 337– 358.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' [13] O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' Ladyˇzenskaja, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' Solonnikov, and N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' Ural’ceva.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' Linear and Quasilinear Equations of Parabolic Type.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' Translations of mathematical monographs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' Providence, RI: American Mathematical Society, 1968.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' [14] G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' Lieberman.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' Second Order Parabolic Differential Equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' World Scientific, 1996.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='1142/3 302.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' [15] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' Moser.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' A Sharp Form of an Inequality by N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' Trudinger.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' Indiana University Mathematics Journal 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='11 (1971), pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' 1077–1092.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' issn: 00222518, 19435258.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' url: http://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='jstor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='org/stable/24890183.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' [16] H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' Poincar´e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' Sur l’uniformisation des fonctions analytiques.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' Acta Mathematica 31 (1908), pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' 1–64.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' [17] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' Schauder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' Der Fixpunktsatz in Funktionalra¨umen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' Studia Mathematica 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='1 (1930), pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' 171–180.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' url: http://eudml.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='org/doc/217247.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' [18] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' Struwe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' A flow approach to Nirenberg’s problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' Duke Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' 128.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='1 (2005), pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' 19–64.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='121 5/S0012-7094-04-12812-X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' [19] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' Struwe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' “Bubbling” of the prescribed curvature flow on the torus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' Journal of the European Mathematical Society 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='10 (2020), pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' 3223–3262.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' [20] N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' Trudinger.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' On embeddings into Orlicz spaces and some applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' Mech.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' 17 (1967), pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' 473–484.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'}