Uniqueness of critical points of solutions to the mean curvature equation with Neumann and Robin boundary conditions

2017/12/22 by Deng, Haiyun, Liu, Hairong, Tian, Long
#35B38 #35J25 #35J93 #Analysis of PDEs (math.AP) #FOS: Mathematics

paper · doi:10.48550/arxiv.1712.08454

Abstract

In this paper, we investigate the critical points of solutions to the prescribed constant mean curvature equation with Neumann and Robin boundary conditions respectively in a bounded smooth convex domain Ω of ℝⁿ(n≥2). Firstly, we show the non-degeneracy and uniqueness of the critical points of solutions in a planar domain by using the local Chen & Huang's comparison technique and the geometric properties of approximate surfaces at the non-degenerate critical points. Secondly, we deduce the uniqueness and non-degeneracy of the critical points of solutions in a rotationally symmetric domain of ℝⁿ(n≥3) by the projection of higher dimensional space onto two dimensional plane.

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