Critical points of solutions for mean curvature equation in strictly convex and nonconvex domains

2017/12/22 by Deng, Haiyun, Liu, Hairong, Tian, Long
#35B38 #35J93 #53C44 #Analysis of PDEs (math.AP) #FOS: Mathematics

paper · doi:10.48550/arxiv.1712.08431

Abstract

In this paper, we mainly investigate the set of critical points associated to solutions of mean curvature equation with zero Dirichlet boundary condition in a strictly convex domain and a nonconvex domain respectively. Firstly, we deduce that mean curvature equation has exactly one nondegenerate critical point in a smooth, bounded and strictly convex domain of ℝⁿ(n≥2). Secondly, we study the geometric structure about the critical set K of solutions u for the constant mean curvature equation in a concentric (respectively an eccentric) spherical annulus domain of ℝⁿ(n≥3), and deduce that K exists (respectively does not exist) a rotationally symmetric critical closed surface S. In fact, in an eccentric spherical annulus domain, K is made up of finitely many isolated critical points (p₁,p₂,⋯,p_l) on an axis and finitely many rotationally symmetric critical Jordan curves (C₁,C₂,⋯,C_k) with respect to an axis.

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