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

TL;DR

This paper investigates the critical points of solutions to the mean curvature equation in convex and nonconvex domains, revealing unique critical points in convex domains and complex structures in spherical annuli.

Contribution

It establishes the uniqueness of a nondegenerate critical point in convex domains and characterizes the geometric structure of critical sets in spherical annuli.

Findings

Exactly one nondegenerate critical point in convex domains

Existence of symmetric critical surfaces in concentric spherical annuli

Finitely many isolated critical points and Jordan curves in eccentric spherical annuli

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 $\mathbb{R}^{n}(n\geq2)$. 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 $\mathbb{R}^{n}(n\geq3)$, 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 ($p_1,p_2,\cdots,p_l$) on an axis and finitely many rotationally symmetric critical Jordan curves ($C_1,C_2,\cdots,C_k$) with respect to an axis.