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

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 $\Omega$ of $\mathbb{R}^{n}(n\geq2)$. 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 $\mathbb{R}^{n}(n\geq3)$ by the projection of higher dimensional space onto two dimensional plane.