diff --git "a/DdAyT4oBgHgl3EQfSPd1/content/tmp_files/load_file.txt" "b/DdAyT4oBgHgl3EQfSPd1/content/tmp_files/load_file.txt" new file mode 100644--- /dev/null +++ "b/DdAyT4oBgHgl3EQfSPd1/content/tmp_files/load_file.txt" @@ -0,0 +1,558 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf,len=557 +page_content='SURFACES OF MINIMUM CURVATURE VARIATION LUIS A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'} +page_content=' CAFFARELLI, PABLO RA´UL STINGA, AND HERN´AN VIVAS Abstract.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'} +page_content=' We establish the analytical theory of surfaces of minimum curvature variation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'} +page_content=' We construct classical, G2 continuous surfaces, as well as weak solutions in the context of geometric measure theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'} +page_content=' Introduction Computer-aided design (CAD) and computer-aided manufacturing (CAM) are widely pop- ular techniques whose basic feature is the use of computer software to create or modify shapes in such a way that some aspects of the design process, such as quality of the object or produc- tivity of the process, are optimized, see, for example, [9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'} +page_content=' Their origins can be traced back to the 1950s and 60s and their development have been continuous since then.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'} +page_content=' Nowadays, CAD/CAM are used in contexts as varied as engineering, particularly in automotive, ship- building and aerospace industries;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'} +page_content=' architectural design;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'} +page_content=' and computer animation for creation of special effects in movies, among many other applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'} +page_content=' Within this realm, of particular interest are geometric problems in computer-aided geomet- ric design (CAGD).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'} +page_content=' The goal of CAGD is the creation of complex smoothly shaped models and surfaces with specified geometric constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'} +page_content=' The resulting surfaces have to accurately reflect these specifications and be free of unwanted wrinkles, bulges and ripples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'} +page_content=' In many instances, the aim is to create fair surfaces that are aesthetically pleasing to the eye.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'} +page_content=' As it turns out, many of these problems can be approached via a variational principle, that is, by looking for a surface that minimizes an appropriate functional or fairness energy subject to adequate geometric boundary conditions, see [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'} +page_content=' The most commonly used fairness energy functionals can be split into two groups: physical- based or geometric-based.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'} +page_content=' The first group roughly corresponds to interpreting the surface as an ideal elastic membrane or plate and minimize energies such as ´ |∇u|2 dx or ´ |∆u|2 dx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'} +page_content=' The second group aims at minimizing energies that relate to geometric invariants of the surface such as the area or curvature, see [11] and the references therein.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'} +page_content=' In 1992, Moreton and S´equin proposed in [8] a numerical algorithm for the creation of 2-dimensional fair surfaces M as minimizers of the energy functional ˆ M ��dκ1 de1 �2 + �dκ2 de2 �2� dA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'} +page_content=' Here e1 and e2 are the principal directions corresponding to the principal curvatures κ1 and κ2 of M and dA is the differential of surface area.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'} +page_content=' It is a key aspect in CAGD to be able to construct fair surfaces that preserve several degrees of geometric continuity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'} +page_content=' This is particularly important at the boundary of the domains where 2010 Mathematics Subject Classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'} +page_content=' Primary: 35B65, 49Q10, 53A10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'} +page_content=' Secondary: 49Q20, 65D17, 68U07.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'} +page_content=' Key words and phrases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'} +page_content=' Curvature variation, computer-aided design, prescribed mean curvature, regularity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'} +page_content=' Research partially supported by NSF grant 1500871 (USA), Simons Foundation grant 580911 (USA), and Agencia Nacional de Promoci´on Cient´ıfica y Tecnol´ogica under grant PICT 2019-3530 (Argentina).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'} +page_content=' 1 arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'} +page_content='00082v1 [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'} +page_content='DG] 31 Dec 2022 2 L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'} +page_content=' CAFFARELLI, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'} +page_content=' STINGA, AND H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'} +page_content=' VIVAS the surfaces meet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'} +page_content=' The notions of geometric continuity are referred to as G0 continuity, where two surfaces meet in a continuous fashion, without jumps;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'} +page_content=' G1 continuity, where the tangent planes of the surfaces meet with continuity;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'} +page_content=' and G2 continuity, where the curvatures meet with continuity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'} +page_content=' These are not the same as the classical notions of C0, C1 and C2 continuities, as those require some specific combination of the derivatives of the solutions to be continuous up to the boundary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'} +page_content=' In particular, G2 continuity turns out to be crucial in applications such as the streamlined surfaces of aircrafts, ships and cars, and this was the main motivation for the numerical study in [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'} +page_content=' In [11], a numerical finite difference method is proposed to construct surfaces that would enjoy G2 continuity as steady states of a sixth order flow derived from the Euler–Lagrange equation of the energy functional ˆ M |∇H|2 dA where H is the mean curvature of M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'} +page_content=' Numerical evidence of G2 continuity is observed in [11], while G1 continuity is expected according to [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'} +page_content=' To the best of our knowledge, the analytical theory of surfaces of minimum curvature variation in general is missing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'} +page_content=' Furthermore, no proof of G2 continuity is available thus far.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'} +page_content=' The aim of this paper is to fill these gaps and to develop the analytical foundation from the PDE perspective of the theory of surfaces of minimum curvature variation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'} +page_content=' We give two constructions of surfaces: classical solutions that are G2-continuous, and weak solutions through geometric measure theory methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'} +page_content=' Therefore, we consider the minimization problem (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'} +page_content='1) min M 1 2 ˆ M |∇MH|2 dA where M ranges over all n−dimensional manifolds in Rn+1, n ≥ 1, with prescribed bound- ary, H is the mean curvature of M and dA is the differential of surface area.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'} +page_content=' Notice that (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'} +page_content='1) minimizes the (quadratic) variation of the mean curvature of M so that surfaces with constant mean curvature such as planes, circles and cylinders are minimizers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'} +page_content=' If M is the graph of a function defined on a bounded domain Ω ⊂ Rn, that is, M = {(x, u(x)) : x ∈ Ω} for some u : Ω → R, then the values of u at ∂Ω prescribe the boundary ∂M of M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'} +page_content=' For a point x0 ∈ Ω, the tangent plane to M at (x0, u(x0)) and its upward pointing unit normal are P(x) = u(x0) + ∇u(x0) · (x − x0) and ν(x0) = (−∇u(x0), 1) (1 + |∇u(x0)|2)1/2 , respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'} +page_content=' The mean curvature H of M at a point is defined as the average of the n principal curvatures of M at that point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'} +page_content=' In the coordinates given by u, it takes the form H ≡ H(u) = 1 n div � ∇u (1 + |∇u|2)1/2 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'} +page_content=' If we set D(u) := (1 + |∇u|2)1/2 we then have that dA = D(u) dx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'} +page_content=' Let f be a function in C1(Ω × R).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'} +page_content=' The tangential gradient of f on M is obtained by projecting the gradient of f in Rn+1 onto the plane orthogonal to ν: ∇Mf = ∇Rn+1f − (ν · ∇Rn+1f)ν on M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'} +page_content=' SURFACES OF MINIMUM CURVATURE VARIATION 3 Clearly, ν · ∇Mf = 0 and (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'} +page_content='2) |∇Mf|2 = |∇Rn+1f|2 − |ν · ∇Rn+1f|2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'} +page_content=' Furthermore, ∇Mf depends only on the values of f on M, see, for instance, [6, Section 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'} +page_content='1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'} +page_content=' To compute ∇MH we extend H as a function of (x, xn+1) ∈ Ω × R by making it constant in xn+1: H(x, xn+1) ≡ H(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'} +page_content=' This is enough to compute ∇MH because the resulting value is independent of the extension.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'} +page_content=' Since ∇Rn+1H = (∇H, Hxn+1) = (∇H, 0), by (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'} +page_content='2) we get |∇MH|2 = |(∇H, 0)|2 − ����(−∇u · ∇H)(−∇u, 1) D(u)2 ���� 2 = |∇H|2 − ���� ∇u · ∇H D(u) ���� 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'} +page_content=' With this formula the energy in (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'} +page_content='1) becomes (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'} +page_content='3) E[M] = 1 2 ˆ Ω � |∇H|2 − ���� ∇u · ∇H D(u) ���� 2� D(u) dx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'} +page_content=' We will call this the geometric energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'} +page_content=' It follows from the Cauchy–Schwartz inequality that |∇H|2 D(u)2 ≤ |∇MH|2 ≤ |∇H|2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'} +page_content=' Therefore, we will also study the (larger) simplified energy functional (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'} +page_content='4) E[H, u] := 1 2 ˆ Ω |∇H|2D(u) dx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'} +page_content=' In Section 2 we consider (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'} +page_content='4) and show how to construct smooth solutions that satisfy the prescribed mean curvature equation for a curvature of minimum variation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'} +page_content=' Section 3 shows how to modify the argument to construct solutions of the geometric energy functional (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'} +page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'} +page_content=' Finally, in Section 4 we provide a weak formulation of the problem and prove existence of minimizers in the context of geometric measure theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'} +page_content=' Existence of G2 surfaces for the simplified energy In this section we work with the simplified energy functional (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'} +page_content='4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'} +page_content=' Let Ω ⊂ Rn be a bounded domain such that ∂Ω ∈ C3,α for some 0 < α < 1 fixed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'} +page_content=' We assume that we are given prescribed boundary values g ∈ C3,α(Ω) for u and h ∈ C1,α(Ω) for H on ∂Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'} +page_content=' We address the following problem: given Ω and the boundary datum g, find a surface given by the graph of a function u such that its mean curvature H is a minimizer of (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'} +page_content='4) among all functions with prescribed boundary values h ∈ C1,α(Ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'} +page_content=' We will use Schauder’s fixed point theorem: Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'} +page_content='1 (see [6, Corollary 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'} +page_content='2]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'} +page_content=' Let G be a closed convex set in a Banach space B and let T be a continuous mapping of G into itself such that the image T(G) is precompact.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'} +page_content=' Then T has a ���xed point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'} +page_content=' Consider the Banach space B = C1,α(Ω) and its subset G := � v ∈ C1,α(Ω) : v = g on ∂Ω � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'} +page_content=' Observe that G is nonempty because g ∈ C3,α(Ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'} +page_content=' By classical global Schauder estimates, we see that another example of function in G is the harmonic extension v of g inside of Ω: � ∆v = 0 in Ω v = g on ∂Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'} +page_content=' 4 L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'} +page_content=' CAFFARELLI, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'} +page_content=' STINGA, AND H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'} +page_content=' VIVAS It is clear that G is convex and closed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'} +page_content=' For any v ∈ G, we define the functional (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'} +page_content='1) E[H, v] := 1 2 ˆ Ω |∇H|2D(v) dx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'} +page_content=' The map T : G → G is constructed in a 2-step process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'} +page_content=' Step 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'} +page_content=' Given any v ∈ G, we find the unique minimizer H ∈ W 1,2(Ω) to (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'} +page_content='1) such that H − h ∈ W 1,2 0 (Ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'} +page_content=' This can be done because the coefficient D(v) satisfies 1 ≤ D(v) ≤ (1 + ∥∇v∥2 L∞(Ω))1/2 < ∞, so that (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'} +page_content='1) is a coercive functional.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'} +page_content=' Then H is the unique weak solution to � div(D(v)∇H) = 0 in Ω H = h on ∂Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'} +page_content=' Since v ∈ C1,α(Ω), the coefficient D(v) ∈ C0,α(Ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'} +page_content=' Thus, by global Schauder estimates (see [6, Section 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'} +page_content='11]), (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'} +page_content='2) ∥H∥C1,α(Ω) ≤ Cn[∂Ω]C1,α∥D(v)∥C0,α(Ω)∥h∥C1,α(∂Ω) where Cn > 0 is a constant that depends only on dimension n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'} +page_content=' Step 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'} +page_content=' Given H ∈ C1,α(Ω) from Step 1, we find the solution u to the prescribed mean curvature equation (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'} +page_content='3) � � � div � ∇u D(u) � = nH in Ω u = g on ∂Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'} +page_content=' For this, we use the following result (where we use nH instead of H in [7]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'} +page_content=' Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'} +page_content='2 ([7, Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'} +page_content='1] and its proof).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'} +page_content=' Let 0 < α < 1 and Ω ⊂ Rn be a bounded domain with C3,α boundary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'} +page_content=' Suppose that H ∈ C1,α(Ω) satisfies (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'} +page_content='4) ∥H∥Ln(Ω) < �ˆ Rn(1 + |p|2)− n+2 2 dp �1/n and, for any y ∈ ∂Ω, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'} +page_content='5) |H(y)| ≤ H∂Ω(y), where H∂Ω is the mean curvature of ∂Ω corresponding to the inner unit normal vector to ∂Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'} +page_content=' Then for any g ∈ C3,α(Ω) there exists a unique solution u ∈ C3,α(Ω) to (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'} +page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'} +page_content=' In particular, there exists a constant C∗ > 0, depending only on n, α, ∥H∥Ln(Ω), ∥H∥C1(Ω), ∥g∥C2,α(Ω) and Ω, such that (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'} +page_content='6) ∥u∥C2,α(Ω) ≤ C∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'} +page_content=' The constant in the right-hand side of (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'} +page_content='4) can be simplified.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'} +page_content=' Recall the definition of the Beta function and its relation with the Gamma function: for x, y > 0, B(x, y) := ˆ ∞ 0 tx−1 (1 + t)x+y dt = Γ(x)Γ(y) Γ(x + y) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'} +page_content=' SURFACES OF MINIMUM CURVATURE VARIATION 5 We have that Γ(1) = 1 and xΓ(x) = Γ(x + 1), for all x > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'} +page_content=' By passing to polar coordinates p = rθ, for r > 0 and θ ∈ Sn−1, performing the change of variables t = r2 which makes 2dr/r = dt/t, and using that |Sn−1| = n|B1|, we get ˆ Rn(1 + |p|2)− n+2 2 dp = |Sn−1| ˆ ∞ 0 rn (1 + r2) n+2 2 dr r = n|B1| 2 ˆ ∞ 0 tn/2 (1 + t) n+2 2 dt t = n|B1| 2 B(n/2, 1) = n|B1| 2 Γ(n/2) Γ(n/2 + 1) = |B1|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'} +page_content=' Therefore, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'} +page_content='4) reads (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'} +page_content='7) ∥H∥Ln(Ω) < |B1|1/n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'} +page_content=' Now (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'} +page_content='5) and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'} +page_content='7) impose further restrictions on the boundary values h of H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'} +page_content=' Condition (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'} +page_content='5) is natural to assume and cannot be avoided (see, for example, the nonexistence results [6, Theorem 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'} +page_content='11] and [7, Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'} +page_content='5]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'} +page_content=' Therefore, we assume that h ∈ C1,α(Ω) additionally satisfies (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'} +page_content='8) |h(y)| ≤ H∂Ω(y) for all y ∈ ∂Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'} +page_content=' On the other hand, by the maximum principle (see [6, Section 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'} +page_content='1]), we can estimate (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'} +page_content='9) ˆ Ω |H|n dx ≤ |Ω| � max ∂Ω |h| �n .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'} +page_content=' or ∥H∥Ln(Ω) ≤ |Ω|1/n max ∂Ω |h|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'} +page_content=' Thus, in order to ensure (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'} +page_content='7), we further assume that (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'} +page_content='10) max ∂Ω |h| < �|B1| |Ω| �1/n .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'} +page_content=' From the computer science point of view, this means that for large boundary curvatures h the domain Ω for the reconstruction should be sufficiently small.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'} +page_content=' Therefore, under the additional assumptions (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'} +page_content='8) and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'} +page_content='10), we can apply Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'} +page_content='2 and find the unique solution u ∈ C3,α(Ω) to (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'} +page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'} +page_content=' This completes Step 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'} +page_content=' Using these two steps, we define T : G → G by T(v) = u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'} +page_content=' In order to apply Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'} +page_content='1, we need to verify that (1) T is continuous, and (2) T(G) is precompact.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'} +page_content=' Let us begin with (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'} +page_content=' Fix v1 ∈ G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'} +page_content=' We need to show that given any ε > 0 there exists δ = δ(ε, v1) > 0 such that for any v2 ∈ G satisfying ∥v1 − v2∥C1,α(Ω) < δ we have ∥u1 − u2∥C1,α(Ω) < ε, where uj = Tvj, for j = 1, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'} +page_content=' Let Hj denote the minimizer of E[·, vj], j = 1, 2, as constructed in Step 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'} +page_content=' Then the difference H = H1 − H2 is the unique weak solution to � div(D(v1)∇H) = div � (D(v2) − D(v1))∇H2 � in Ω H = 0 on ∂Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'} +page_content=' 6 L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'} +page_content=' CAFFARELLI, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'} +page_content=' STINGA, AND H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'} +page_content=' VIVAS By global Schauder estimates (see [6, Section 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'} +page_content='11]), (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'} +page_content='11) ∥H∥C1,α(Ω) ≤ Cn[∂Ω]C1,α∥D(v1)∥C0,α(Ω)∥(D(v2) − D(v1))∇H2∥C0,α(Ω) ≤ C(n, α, Ω, v1, ∇H2)∥v1 − v2∥C1,α(Ω) =: C1∥v1 − v2∥C1,α(Ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'} +page_content=' Let us now estimate the difference u = u1 − u2 ∈ C3,α(Ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'} +page_content=' Since � � � div � ∇ui D(ui) � = nHi in Ω, for i = 1, 2 u1 = u2 = g on ∂Ω we find that � � � div � ∇u1 D(u1) − ∇u2 D(u2) � = nH in Ω u = 0 on ∂Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'} +page_content=' In order to apply global Schauder estimates one more time we need to find an equation for u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'} +page_content=' Set F(p) := p � 1 + |p|2 p ∈ Rn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'} +page_content=' Then F is a smooth, bounded vector field with entries Fi(p) = pi √ 1+|p|2 , for i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'} +page_content=' , n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'} +page_content=' Note that, for j = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'} +page_content=' , n, ∂jFi(p) = � � � 1 √ 1+|p|2 − p2 i (1+|p|2)3/2 if i = j − pipj (1+|p|2)3/2 if i ̸= j = δij D(p) − pipj D(p)3 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'} +page_content=' In particular, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'} +page_content='12) ∂jFi(p) = ∂iFj(p) so that ∇F is a symmetric matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'} +page_content=' It is clear that ∇F is bounded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'} +page_content=' To see that it is locally strictly elliptic, observe that, for any ξ ∈ Rn, by the Cauchy–Schwartz inequality, n � i,j=1 ∂jFi(p)ξiξj = n � i,j=1 �δijξiξj D(p) − pipjξiξj D(p)3 � ≥ |ξ|2 � 1 D(p) − |p|2 D(p)3 � = |ξ|2 D(p)3 ≥ θ(R)|ξ|2 for all |p| < R, where θ(R) → 0 as R → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'} +page_content=' Furthermore, we can write Fi(∇u1) − Fi(∇u2) = ˆ 1 0 d dtFi(t∇u1 + (1 − t)∇u2) dt = ˆ 1 0 ∇Fi(t∇u1 + (1 − t)∇u2) · ∇(u1 − u2) dt so that F(∇u1) − F(∇u2) = A(x)∇u with Aij(x) = ˆ 1 0 ∂jFi(t∇u1 + (1 − t)∇u2) dt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'} +page_content=' SURFACES OF MINIMUM CURVATURE VARIATION 7 The matrix A is symmetric thanks to (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'} +page_content='12), as well as bounded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'} +page_content=' Recall that ∇F is locally strictly elliptic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'} +page_content=' Now, u1 ∈ C3,α(Ω) is fixed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'} +page_content=' By (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'} +page_content='6), the C2,α(Ω) norm of u2 is uniformly controlled by the C1(Ω) norm of H2, which in turn is uniformly close to the C1(Ω) norm of the initially fixed H1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'} +page_content=' These facts imply that that A(x) is strictly elliptic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'} +page_content=' Moreover, we have the following technical lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'} +page_content=' Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'} +page_content=' Let U, V : Ω → Rn, U, V ∈ C0,α(Ω) and let ψ : Rn → R be a smooth function such that ∥ψ∥L∞(Rn) + ∥∇ψ∥L∞(Rn) < ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'} +page_content=' Define φ(x) := ˆ 1 0 ψ(tU(x) + (1 − t)V (x)) dt for every x ∈ Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'} +page_content=' Then φ ∈ C0,α(Ω), with ∥φ∥C0,α(Ω) ≤ ∥ψ∥L∞(Rn) + ∥∇ψ∥L∞(Rn) � [U]Cα(Ω) + [V ]Cα(Ω) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'} +page_content=' The boundedness of ψ implies that φ is bounded with ∥φ∥L∞(Ω) ≤ ∥ψ∥L∞(Rn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'} +page_content=' To bound the H¨older seminorm of φ, let x, y ∈ Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'} +page_content=' Then |φ(x) − φ(y)| = ���� ˆ 1 0 � ψ(tU(x) + (1 − t)V (x)) − ψ(tU(y) + (1 − t)V (y)) � dt ���� ≤ ∥∇ψ∥L∞(Rn) ˆ 1 0 |t(U(x) − U(y)) + (1 − t)(V (x) − V (y))| dt ≤ ∥∇ψ∥L∞(Rn) (|U(x) − U(y)| + |V (x) − V (y)|) ≤ ∥∇ψ∥L∞(Rn) � [U]Cα(Ω) + [V ]Cα(Ω) � |x − y|α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'} +page_content=' □ Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'} +page_content='3 gives the H¨older continuity of A(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'} +page_content=' Indeed, let ψ be any of the entries of the gradient matrix of F: (∇F(p))ij = δij D(p) − pipj D(p)3 for i, j = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'} +page_content=' , n, which are smooth and bounded, so ∥ψ∥L∞(Rn) ≤ M1, where M1 is independent of i and j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'} +page_content=' For any k = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'} +page_content=' , n, we have ∂k(∇F(p))ij = −δijpk + δikpj + δjkpi D(p)3 + pipjpk D(p)5 and these are all bounded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'} +page_content=' Therefore, ∥∇ψ∥L∞(Rn) ≤ M2, where M2 > 0 is independent of i and j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'} +page_content=' By setting U = ∇u1 and V = ∇u2 in Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'} +page_content='3, we get ∥A∥C0,α(Ω) ≤ M1 + M2 � [∇u1]Cα(Ω) + ([∇u2]Cα(Ω) � ≤ M3 with M3 > 0 a constant depending only on n, α, ∥H1∥Ln(Ω), ∥H1∥C1(Ω), ∥g∥C2,α(Ω), and Ω, see (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'} +page_content='6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'} +page_content=' Observe that all these quantities are independent of u2 if v2 is close to v1 in C1,α(Ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'} +page_content=' In summary, we have found that u is a solution to � div(A(x)∇u) = nH in Ω u = 0 on ∂Ω and so, by Schauder estimates, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'} +page_content='13) ∥u∥C1,α(Ω) ≤ Cn[∂Ω]C1,αM3∥H∥C1,α(Ω) =: C2∥H∥C1,α(Ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'} +page_content=' 8 L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'} +page_content=' CAFFARELLI, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'} +page_content=' STINGA, AND H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'} +page_content=' VIVAS Therefore, by collecting estimates (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'} +page_content='11) and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'} +page_content='13), and recalling that u = u1 − u2 = Tv1 − Tv2, and H = H1 − H2, we obtain ∥Tv1 − Tv2∥C1,α(Ω) ≤ C1C2∥v1 − v2∥C1,α(Ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'} +page_content=' If we choose δ = ε/(C1C2) then we see that T is continuous, as desired.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'} +page_content=' Let us now turn to (2), which will follow from a priori estimates for prescribed mean curvature equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'} +page_content=' Let {vk}k≥1 be a sequence in G such that sup k≥1 ∥vk∥C1,α(Ω) ≤ N1 < ∞ and consider the corresponding solutions Hk ∈ C1,α(Ω) found in Step 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'} +page_content=' Set uk = Tvk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'} +page_content=' By (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'} +page_content='6), ∥uk∥C2,α(Ω) ≤ Ck where Ck > 0 is a constant depending only on n, α, ∥Hk∥Ln(Ω), ∥Hk∥C1(Ω), ∥h∥C2,α(Ω), and Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'} +page_content=' Since all Hk have the same boundary values h, by (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'} +page_content='9), we get that sup k≥1 ∥Hk∥Ln(Ω) = N2 < ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'} +page_content=' Furthermore, from the C1,α estimate in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'} +page_content='2), sup k≥1 ∥Hk∥C1(Ω) ≤ Cn[∂Ω]C1,α∥h∥C1,α(∂Ω) sup k≥1 ∥D(vk)∥C0,α(Ω) = N3 < ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'} +page_content=' Consequently, sup k≥1 ∥uk∥C2,α(Ω) ≤ sup k≥1 Ck = N4 < ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'} +page_content=' By the Arzel`a–Ascoli compact embedding C2,α(Ω) ⊂⊂ C1,α(Ω), there exist a subsequence {ukj}j≥1 of {uk}k≥1 and u ∈ G such that ukj → u in C1,α(Ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'} +page_content=' We conclude that T(G) is precompact and (2) is proved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'} +page_content=' Thus, by Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'} +page_content='1, there exists u ∈ G such that Tu = u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'} +page_content=' We have proved the following: Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'} +page_content='4 (Existence for the simplified energy).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'} +page_content=' Let Ω ⊂ Rn be a bounded domain with C3,α boundary ∂Ω, for some 0 < α < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'} +page_content=' Fix g ∈ C3,α(Ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'} +page_content=' Let h ∈ C1,α(Ω) such that (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'} +page_content='14) |h(y)| ≤ H∂Ω(y) for all y ∈ ∂Ω, where H∂Ω is the mean curvature of ∂Ω corresponding to the inner unit normal vector to ∂Ω, and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'} +page_content='15) max ∂Ω |h| < �|B1| |Ω| �1/n .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'} +page_content=' Then there exist u ∈ C3,α(Ω) and H ∈ C1,α(Ω) such that H minimizes the energy 1 2 ˆ Ω |∇H|2D(u) dx among all H ∈ W 1,2(Ω) such that H − h ∈ W 1,2 0 (Ω), or, equivalently, H is the unique weak solution to � div(D(u)∇H) = 0 in Ω H = h on ∂Ω, SURFACES OF MINIMUM CURVATURE VARIATION 9 and, in addition, H is the mean curvature of the graph of u with prescribed values on ∂Ω, that is, � � � 1 n div � ∇u D(u) � = H in Ω u = g on ∂Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'} +page_content=' Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'} +page_content='5 (Nonexistence of solutions).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'} +page_content=' The conditions imposed on the curvature at the boundary datum h in Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'} +page_content='4 come from restrictions already present when one seeks for solutions of the prescribed mean curvature equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'} +page_content=' Indeed, the divergence form equation for H is uniformly elliptic when u is, say, Lipschitz continuous and therefore is always solvable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'} +page_content=' On the other hand, if condition (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'} +page_content='14) is not satisfied, that is, |h(y0)| > H∂Ω(y0) for some y0 ∈ ∂Ω and h ≥ 0 (or h ≤ 0) on ∂Ω then H ≥ 0 (or H ≤ 0) in Ω and we have that for any ε > 0 there exists g ∈ C∞(Ω) with |g| < ε such that the prescribed mean curvature equation with curvature H and boundary values h is not solvable (see [7, Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'} +page_content='5] or [6, Corollary 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'} +page_content='13]) and hence neither is the minimum curvature variation system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'} +page_content=' On the other hand, a necessary condition for existence of solutions of the prescribed mean curvature equation is (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'} +page_content='16) ���� ˆ Ω Hη dx ���� ≤ 1 − ε0 n ˆ Ω |∇η| dx for all η ∈ C1 0(Ω) and with 1 − ε0 = sup Ω |∇u| � 1 + |∇u|2 , see [6, eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'} +page_content=' (16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'} +page_content='60)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'} +page_content=' This condition implies ∥H∥Ln(Ω) < |B1|1/n, which is the structural con- dition on H that motivates (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'} +page_content='15).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'} +page_content=' The requirement in Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'} +page_content='4 could be thus weakened, but (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'} +page_content='16) is the least requirement under which existence for the prescribed mean curvature equation can be obtained and hence also for the system at hand.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'} +page_content=' Existence of G2 surfaces for the geometric energy In this section we discuss how the technique we developed in the previous section can be applied to the geometric energy functional E[M] = 1 2 ˆ Ω � |∇H|2 − ���� ∇u · ∇H D(u) ���� 2� D(u) dx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'} +page_content=' Let Ω, α, h and g be as in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'} +page_content=' Fix v ∈ C1,α(Ω) such that v = g on ∂Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'} +page_content=' Consider the energy Ev[H] := 1 2 ˆ Ω � |∇H|2 − ���� ∇v · ∇H D(v) ���� 2� D(v) dx = ˆ Ω L(∇H) dx where the smooth Lagrangian L is given by L(p) = 1 2 � |p|2 − ���� ∇v · p D(v) ���� 2� D(v) for p ∈ Rn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'} +page_content=' Then L is coercive, as L(p) ≥ 1 2 � |p|2 − |∇v|2|p|2 D(v)2 � D(v) = 1 2 � D(v) − |∇v|2 D(v) � |p|2 = 1 2D(v)|p|2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'} +page_content=' 10 L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'} +page_content=' CAFFARELLI, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'} +page_content=' STINGA, AND H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'} +page_content=' VIVAS To prove that L is convex, first observe that, for i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'} +page_content=' , n, Lpi(p) = � pi − (∇v · p) D(v)2 vxi � D(v) = n � j=1 � δijD(v) − vxivxj D(v) � pj and, for i, j = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'} +page_content=' , n, Lpipj(p) = δijD(v) − vxivxj D(v) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'} +page_content=' Then, for any ξ ∈ Rn, Lpipj(p)ξiξj = D(v)|ξ|2 − (∇v · ξ)2 D(v) ≥ � D(v) − |∇v|2 D(v) � |ξ|2 = 1 D(v)|ξ|2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'} +page_content=' Thus, D2 pL is a positive definite matrix, and L is uniformly convex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'} +page_content=' It follows that there exists a unique minimizer H ∈ W 1,2(Ω) of the energy Ev[H] such that H − h ∈ W 1,2 0 (Ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'} +page_content=' In particular, H is the unique weak solution to � � � � � n � i=1 (Lpi(∇H))xi = 0 in Ω H = h on ∂Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'} +page_content=' Since Lpi(∇H) = n � j=1 � δijD(v) − vxivxj D(v) � Hxj we find that H is the unique weak solution to the linear problem � div(a(x)∇H) = 0 in Ω H = h on ∂Ω where aij(x) = δijD(v) − vxivxj D(v) = Lpipj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'} +page_content=' Observe that |aij(x)| ≤ C � D(v) + |∇v|2 D(v) � ≤ C(D(v) + |∇v|) ≤ C(n, ∥∇v∥L∞(Ω)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'} +page_content=' We have already seen that aij(x) is uniformly elliptic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'} +page_content=' Moreover, if v ∈ C1,α(Ω) then aij(x) ∈ C0,α(Ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'} +page_content=' Hence, H ∈ C1,α(Ω), with ∥H∥C1,α(Ω) ≤ Cn[∂Ω]C1,α∥v∥C1,α(Ω)∥h∥C1,α(∂Ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'} +page_content=' If h satisfies (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'} +page_content='8) and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'} +page_content='10) then we can apply Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'} +page_content='2 and find the unique solution u ∈ C3,α(Ω) to (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'} +page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'} +page_content=' From here on we can continue with the fixed point arguments we did in Section 2 to conclude the following result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'} +page_content=' Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'} +page_content='1 (Existence for the geometric functional).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'} +page_content=' Let Ω ⊂ Rn be a bounded domain with C3,α boundary ∂Ω, for some 0 < α < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'} +page_content=' Fix g ∈ C3,α(Ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'} +page_content=' Let h ∈ C1,α(Ω) such that |h(y)| ≤ H∂Ω(y) for all y ∈ ∂Ω, SURFACES OF MINIMUM CURVATURE VARIATION 11 where H∂Ω is the mean curvature of ∂Ω corresponding to the inner unit normal vector to ∂Ω, and max ∂Ω |h| < �|B1| |Ω| �1/n .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'} +page_content=' Then there exist u ∈ C3,α(Ω) and H ∈ C1,α(Ω) such that H minimizes the energy 1 2 ˆ Ω � |∇H|2 − ���� ∇u · ∇H D(u) ���� 2� D(u) dx among all H ∈ W 1,2(Ω) such that H − h ∈ W 1,2 0 (Ω), or, equivalently, H is the unique weak solution to � div(a(x)∇H) = 0 in Ω H = h on ∂Ω, where aij(x) = δijD(u) − uxiuxj D(u) and, in addition, H is the mean curvature of the graph of u with prescribed values on ∂Ω, that is, � � � 1 n div � ∇u D(u) � = H in Ω u = g on ∂Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'} +page_content=' Weak solutions In this section we develop the weak formulation of the minimum curvature variation prob- lem in the context of geometric measure theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'} +page_content=' Given a Lipschitz bounded domain Ω, we denote by BV(Ω) the space of functions of bounded variation in Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'} +page_content=' We start by recalling that u ∈ BV(Ω) is a generalized solution to the prescribed mean curvature equation with (weak) mean curvature H ∈ L1(Ω) and boundary value g ∈ L1(∂Ω) if (WPMC) J [u] = min v∈BV(Ω) J [v] where J [v] := ˆ Ω D(v) + ˆ Ω nHv dx + ˆ ∂Ω |v − g| dS and (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'} +page_content='1) ˆ Ω D(v) := sup � ˆ Ω � v n � i=1 ∂xiφi + φn+1 � dx : φi ∈ C1 c (Ω), n+1 � i=1 φ2 i ≤ 1 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'} +page_content=' Note that ´ Ω � 1 + |∇u|2 dx does not make usual sense a priori for a function of bounded variation and so (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'} +page_content='1) is indeed a definition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'} +page_content=' Furthermore, this definition is consistent in the sense that for v ∈ W 1,1(Ω) we have ˆ Ω D(v) = ˆ Ω � 1 + |∇v|2 dx, see the proof of Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'} +page_content=' 12 L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'} +page_content=' CAFFARELLI, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'} +page_content=' STINGA, AND H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'} +page_content=' VIVAS In [3, Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'} +page_content='1], Giaquinta proved that if H is a measurable function then (WPMC) is solvable in BV(Ω) if and only if there exists ε0 > 0 such that, for every measurable subset A ⊂ Ω, (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'} +page_content='2) ���� ˆ A H dx ���� ≤ (1 − ε0) 1 nP(∂A) where P(∂A) denotes the perimeter of A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'} +page_content=' Clearly, (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'} +page_content='2) is significant only when A is a set of finite perimeter (or Caccioppoli set).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'} +page_content=' We need a generalized measure of surface area.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'} +page_content=' In that regard, we recall that the dis- tributional gradient of u ∈ BV(Ω) is a vector valued Radon measure whose total variation is identified with |∇u|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'} +page_content=' This is again consistent in the sense that if u ∈ W 1,1(Ω) then the total variation equals ´ Ω |∇u| dx (see [2, Chapter 5] for this and other properties of the space BV(Ω) used hereafter).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'} +page_content=' In general, for an open set U ⊂⊂ Ω the variation measure of ∇u over U is given by |∇u|(U) = sup �ˆ U u div φ dx : φ ∈ C1 c (U;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'} +page_content=' Rn), |φ| ≤ 1 � and, for an arbitrary set V ⊂ Ω, |∇u|(V ) = inf � |∇u|(U) : V ⊂ U and U is open � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'} +page_content=' Taking this into account, and in analogy with (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'} +page_content='1), we define for the area measure by (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'} +page_content='3) D(u)(U) = sup � ˆ Ω � u n � i=1 ∂xiφi + φn+1 � dx : φi ∈ C1 c (U), n+1 � i=1 φ2 i ≤ 1 � for any U ⊂⊂ Ω open and, for an arbitrary set V ⊂ Ω, D(u)(V ) = inf {D(u)(U) : V ⊂ U and U is open} .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'} +page_content=' Although (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'} +page_content='3) could be defined, in principle, for functions in L1(Ω), it is easy to check that (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'} +page_content='3) is finite if and only if u ∈ BV(Ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'} +page_content=' Similarly as for the variation measure, D(u) is a Radon measure, namely, a locally finite, Borel regular measure in Rn (to prove that it is locally finite, see the ideas in [5, eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'} +page_content=' (14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'} +page_content='2)]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'} +page_content=' The following observation will be useful.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'} +page_content=' Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'} +page_content=' Let U ⊂ Ω be a Borel set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'} +page_content=' Then (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'} +page_content='4) |U| ≤ D(u)(U).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'} +page_content=' Due to the Borel regularity of both D(u) and the Lebesugue measure it suffices to prove (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'} +page_content='4) for open sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'} +page_content=' Let U ⊂ Rn be open.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'} +page_content=' First, we note that (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'} +page_content='5) D(u)(U) = ˆ U � 1 + |∇u|2 dx for any u ∈ C1(Ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'} +page_content=' Indeed, an integration by parts yields ˆ Ω � u n � i=1 ∂xiφi + φn+1 � dx = ˆ U (−∇u, 1) · Φ dx where Φ = (φ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'} +page_content=' , φn, φn+1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'} +page_content=' Then the Cauchy-Schwartz inequality in Rn+1 and the condi- tion |Φ| ≤ 1 give D(u)(U) ≤ ˆ U � 1 + |∇u|2 dx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'} +page_content=' SURFACES OF MINIMUM CURVATURE VARIATION 13 On the other hand, � 1 + |∇u|2 ∈ L1(U) and so there exists a sequence Φj = (φj 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'} +page_content=' , φj n, φj n+1) with φj i ∈ C1 c (U), j ≥ 1, that converges in L1(U) and almost everywhere to (−∇u,1) √ 1+|∇u|2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'} +page_content=' Fur- thermore, (−∇u,1) √ 1+|∇u|2 is a unit vector so we may assume that �n+1 i=1 (φj i)2 ≤ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'} +page_content=' Since ��Φj · (−∇u, 1) �� ≤ |Φj| � 1 + |∇u|2 ≤ � 1 + |∇u|2 ∈ L1(U) we can use the dominated convergence theorem to get lim j→∞ ˆ Ω � u n � i=1 ∂xiφj i + φj n+1 � dx = lim j→∞ ˆ U (−∇u, 1) · Φj dx = ˆ U � 1 + |∇u|2 dx and the supremum is achieved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'} +page_content=' Thus (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'} +page_content='5) holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'} +page_content=' Second, we have that u ∈ BV(U) and there exists {uk}k≥1 ⊂ BV(U) ∩ C∞(U) such that uk → u in L1(Ω) and (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'} +page_content='6) lim k→∞ D(uk)(U) = D(u)(U), see [2, Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'} +page_content='3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'} +page_content=' Since, by (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'} +page_content='5), the conclusion (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'} +page_content='4) is trivial for C1 functions, we have |U| ≤ lim k→∞ D(uk)(U) = D(u)(U) as desired.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'} +page_content=' □ From now on, we fix a bounded, C1,1 domain Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'} +page_content=' We consider the minimization problem min (u,H)∈A I[u, H] where (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'} +page_content='7) I[u, H] := ˆ Ω |∇H|2 dD(u) and dD(u) stands for the area measure defined in (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'} +page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'} +page_content=' The admissible set A is defined as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'} +page_content=' Let h ∈ W 2,2(Ω) ∩ Lip(∂Ω) satisfying (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'} +page_content='8) |h(y)| ≤ n − 1 n Λ(y), y ∈ ∂Ω, and max ∂Ω |h| ≤ (1 − ε0) �|B1| |Ω| �1/n , where Λ(y) is the weak mean curvature of ∂Ω at y ∈ ∂Ω and (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'} +page_content='9) n − 1 n < ε0 < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'} +page_content=' Define (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'} +page_content='10) A := � (u, H) ∈ BV(Ω) × (Ln(Ω) ∩ W 2,2(Ω)) : u solves (WPMC) and ∥H∥Ln(Ω) + ∥H∥W 2,2(Ω) ≤ C0, H = h on ∂Ω � with C0 > 0 is to be appropriately chosen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'} +page_content=' The equality H = h is understood in the sense of traces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'} +page_content=' Remark 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'} +page_content=' The condition H ∈ W 2,2(Ω) is certainly natural for applications to the design of fair G2-continuous surfaces in CAD/CAM/CAGD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'} +page_content=' Indeed, in dimensions n = 1, 2, 3, the Sobolev embedding gives that the curvature H is H¨older continuous.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'} +page_content=' The main result of this section is the following: Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'} +page_content='3 (Existence of weak solutions).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'} +page_content=' Let I be defined by (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'} +page_content='7), h ∈ W 2,2(Ω)∩Lip(∂Ω) satisfying (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'} +page_content='8) and ε0 ∈ (0, 1) satisfying (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'} +page_content='9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'} +page_content=' Then the set of admissible functions A in (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'} +page_content='10) is nonempty and there exists a minimizer of I within the class A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'} +page_content=' 14 L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'} +page_content=' CAFFARELLI, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'} +page_content=' STINGA, AND H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'} +page_content=' VIVAS To prove Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'} +page_content='3 we recall the notion and properties of Γ−convergence in our context, referring the reader to [1] for an introduction to the topic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'} +page_content=' Let Jk, k ≥ 1, and J∞ be functionals defined on the common space BV(Ω) and taking values in [−∞, ∞].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'} +page_content=' The sequence {Jk}k≥1 is said to Γ−converge to J∞ if the following two conditions hold: (a) For every v ∈ BV(Ω) and every sequence {vk}k≥1 ⊂ BV(Ω) such that vk → v in BV(Ω) it holds lim inf k→∞ Jk(vk) ≥ J∞(v).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'} +page_content=' (b) For every v ∈ BV(Ω) there exists a sequence {vk}k≥1 ⊂ BV(Ω) such that vk → v in BV(Ω) for which lim sup k→∞ Jk(vk) ≤ J∞(v).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'} +page_content=' We will use the following result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'} +page_content=' Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'} +page_content='4 (see [1, Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'} +page_content='21]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'} +page_content=' Let (X, d) be a metric space and let {fk}k≥1 be an equi-mildly coercive sequence of functions on X that Γ−converges to f∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'} +page_content=' Then, there exits min X f∞ = lim k→∞ inf X fk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'} +page_content=' Moreover, if {xk}k≥1 ⊂ X is a precompact sequence such that lim k→∞ fk(xk) = lim k→∞ inf X fk then every limit of {xk}k≥1 is a minimum point for f∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'} +page_content=' Here f is said to be mildly coercive if there exists a nonempty compact set K ⊂ X such that infK f = infX f, and equi-mild coercivity means that the set K is the same for the whole sequence {fk}k≥1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'} +page_content=' Proof of Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'} +page_content=' The proof is divided into 4 steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'} +page_content=' Step 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'} +page_content=' A ̸= ∅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'} +page_content=' We can extend h to Ω by solving � ∆H = 0 in Ω H = h on ∂Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'} +page_content=' By classical elliptic regularity, H ∈ W 2,2(Ω) and ∥H∥W 2,2(Ω) ≤ C0 where C0 = C0(∂Ω, ∥h∥L∞(∂Ω)) > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'} +page_content=' Moreover, by the H¨older and isoperimetric inequalities, ���� ˆ A H dx ���� ≤ ∥H∥Ln(Ω)|A| n−1 n ≤ ∥H∥Ln(Ω) P(∂A) n|B1|1/n .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'} +page_content=' By the maximum principle and (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'} +page_content='8) we have ∥H∥Ln(Ω) ≤ |Ω|1/n max ∂Ω |h| ≤ (1 − ε0)|B1|1/n, where we make C0 larger if needed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'} +page_content=' Therefore, ���� ˆ A H dx ���� ≤ (1 − ε0) n P(∂A) and (WPMC) is solvable for this H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'} +page_content=' Let u ∈ BV(Ω) be the corresponding minimizer of J .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'} +page_content=' We have that A ̸= ∅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'} +page_content=' We further point out that ´ Ω D(u) < ∞ and H ∈ Lip(Ω) so that ˆ Ω |∇H|2 dD(u) ≤ ∥∇H∥2 L∞(Ω)D(u)(Ω) < ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'} +page_content=' SURFACES OF MINIMUM CURVATURE VARIATION 15 In particular, 0 ≤ inf (u,H)∈A I[u, H] < ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'} +page_content=' Step 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'} +page_content=' Construction of a minimizer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'} +page_content=' Let {(uk, Hk)}k≥1 ⊂ A be a minimizing sequence: m := inf (u,H)∈A I[u, H] = lim k→∞ I[uk, Hk].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'} +page_content=' To get a convergent subsequence of {uk}k≥1 we show its uniform boundedness in BV(Ω) and use that BV(Ω) embedds compactly in L1(Ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'} +page_content=' Since every uk is a minimizer of the functional Jk defined by (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'} +page_content='11) Jk[v] := ˆ Ω D(v) + ˆ Ω nHkv dx + ˆ ∂Ω |v − g| dS we have that, for any u0 ∈ BV(Ω), ˆ Ω D(uk) + ˆ Ω nHkuk dx + ˆ ∂Ω |uk − g| dS ≤ ˆ Ω D(u0) + ˆ Ω nHku0 dx + ˆ ∂Ω |u0 − g| dS from where (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'} +page_content='12) ˆ Ω D(uk) + ˆ Ω nHkuk dx ≤ C + ˆ Ω nHku0 dx for C > 0 independent of k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'} +page_content=' Reasoning as in [3, eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'} +page_content=' (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'} +page_content='4)] we have that (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'} +page_content='13) ˆ Ω Hkuk dx ≥ −(1 − ε0) ˆ Ω |∇uk| − C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'} +page_content=' Furthermore, BV(Ω) ⊂ L n n−1 (Ω) so (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'} +page_content='13) and the H¨older inequality in (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'} +page_content='12) give ˆ Ω D(uk) ≤ −n ˆ Ω Hkuk dx + C + ˆ Ω nHku0 dx ≤ n(1 − ε0) ˆ Ω |∇uk| + n∥Hk∥Ln(Ω)∥u0∥L n n−1 (Ω) + C for a new constant C > 0 that is independent of k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'} +page_content=' Moreover, the uniform bound on the Ln(Ω) norm of {Hk}k≥1 (they all belong to A) gives ˆ Ω |∇uk| ≤ n(1 − ε0) ˆ Ω |∇uk| + nC0∥u0∥L n n−1 (Ω) + C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'} +page_content=' Thus, after rearranging terms and recalling (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'} +page_content='9), ˆ Ω |∇uk| ≤ 1 (1 − n(1 − ε0)) � nC0∥u0∥L n n−1 (Ω) + C � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'} +page_content=' Hence, by compactness in BV(Ω), there exists a subsequence of {uk}k≥1, still denoted by the same indexes, and u∞ ∈ BV(Ω) such that uk → u∞ in L1(Ω) as k → ∞, and |∇u∞|(Ω) ≤ lim inf k→∞ |∇uk|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'} +page_content=' Note that we also have (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'} +page_content='14) D(u∞) ≤ lim inf k→∞ D(uk).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'} +page_content=' By Poincar´e’s inequality and the Rellich–Kondrachov compactness theorem, there exist a subsequence of {Hk}k≥1, still denoted by the same indexes, and H∞ ∈ W 2,2(Ω) such that (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'} +page_content='15) ∇Hk → ∇H∞ in L2(Ω), as k → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'} +page_content=' 16 L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'} +page_content=' CAFFARELLI, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'} +page_content=' STINGA, AND H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'} +page_content=' VIVAS Further, due to the uniform bound on ∥Hk∥Ln(Ω), we may assume that Hk converges weakly in Ln(Ω) to H∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'} +page_content=' Finally, the weak convergence ensures that ∥H∞∥Ln(Ω) + ∥H∞∥W 2,2(Ω) ≤ C0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'} +page_content=' Step 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'} +page_content=' (u∞, H∞) ∈ A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'} +page_content=' For this step we use Γ−convergence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'} +page_content=' Recall the functionals Jk defined in (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'} +page_content='11) (for the subsequence Hk we found in Step 2) and define J∞ analogously.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'} +page_content=' We want to show that u∞ is a solution of (WPMC), namely, that u∞ is a minimizer of J∞ over BV(Ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'} +page_content=' Let us show that {Jk}k≥1 Γ−converges to J∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'} +page_content=' A first remark is that it is enough to prove the Γ−convergence of � Jk(v) := ˆ Ω vHk dx to � J∞(v) := ˆ Ω vH∞ dx since the other two terms do not depend on k and can be considered as continuous pertur- bations of Jk, see [1, Remark 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'} +page_content='7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'} +page_content=' To prove the liminf inequality (a), let {vk}k≥1 ⊂ BV(Ω) and v ∈ BV(Ω) such that vk → v in BV(Ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'} +page_content=' We write ˆ Ω vkHk dx − ˆ Ω vH∞ dx = Ik + IIk + IIIk with Ik = ˆ Ω (vk − v)H∞ dx IIk = ˆ Ω (vk − v)(Hk − H∞) dx IIIk = ˆ Ω v(Hk − H∞) dx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'} +page_content=' By lower semicontinuity [4, Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'} +page_content='1], lim inf k→∞ Ik ≥ 0 Next, we bound |IIk| ≤ ∥vk − v∥L n n−1 (Ω) � ∥Hk∥Ln(Ω) + ∥H∞∥Ln(Ω) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'} +page_content=' Since vk converge to v in BV(Ω), by the isoperimetric embedding, the convergence also holds in L n n−1 (Ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'} +page_content=' This and the uniform bound of Hk in Ln(Ω) give lim k→∞ IIk = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'} +page_content=' Finally, limk→∞ IIIk = 0 by the weak convergence of Hk to H∞ in Ln(Ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'} +page_content=' As for the limsup inequality (b), given any v ∈ BV(Ω), consider the constant sequence vk = v for all k ≥ 1 and notice that, using the weak convergence of Hk to H∞ in Ln(Ω), we have that lim k→∞ � Jk(vk) = � J∞(v).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'} +page_content=' Hence, {Jk}k≥1 converges to J∞ in the Γ sense.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'} +page_content=' Therefore, we can apply Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'} +page_content='4 with X = BV(Ω), fk = Jk, f∞ = J∞ and {xk}k≥1 and x∞ given by {uk}k≥1 and u∞, respectively (note that the sequence {Jk}k is equi-mildly coercive), to conclude that u∞ is a minimizer of J∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'} +page_content=' We have thus shown that (u∞, H∞) ∈ A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'} +page_content=' SURFACES OF MINIMUM CURVATURE VARIATION 17 Step 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'} +page_content=' (u∞, H∞) is a minimizer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'} +page_content=' Recall that L(p) = 1 2|p|2, p ∈ Rn, is convex, that is, 1 2|p|2 ≥ 1 2|p0|2 + p0 · (p − p0) for every p, p0 ∈ Rn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'} +page_content=' Then we can write 1 2 ˆ Ω |∇Hk|2 dD(uk) ≥ 1 2 ˆ Ω |∇H∞|2 dD(uk) + ˆ Ω ∇H∞ · (∇Hk − ∇H∞) dD(uk).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'} +page_content=' As k → ∞, the left hand side of this inequality converges to m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'} +page_content=' As for the right hand side, (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'} +page_content='6) implies that lim inf k→∞ 1 2 ˆ Ω |∇H∞|2 dD(uk) ≥ 1 2 ˆ Ω |∇H∞|2 dD(u∞).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'} +page_content=' It remains to analyze the second term on the right hand side.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'} +page_content=' For this, notice that Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'} +page_content='1 implies that dD(uk) is absolutely continuous with respect to the Lebesgue measure, see (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'} +page_content='4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'} +page_content=' This and H¨older’s inequality give ���� ˆ Ω ∇H∞ · (∇Hk − ∇H∞) dD(uk) ���� ≤ ˆ Ω |∇H∞||∇Hk − ∇H∞| dD(uk) ≤ C ˆ Ω |∇H∞||∇Hk − ∇H∞| dx ≤ C∥∇H∞∥L2(Ω)∥∇Hk − ∇H∞∥L2(Ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'} +page_content=' In view of (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'} +page_content='15), this term goes to 0 as k → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'} +page_content=' We have shown that m ≥ ˆ Ω |∇H∞|2 dD(u∞).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'} +page_content=' Since (u∞, H∞) ∈ A equality must be attained and (u∞, H∞) is a minimizer, as desired.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'} +page_content=' □ References [1] A.' metadata={'source': 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