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

Haiyun Deng, Hairong Liu, Long Tian

Published 2017-12-22Version 1

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 smooth, bounded and strictly convex domain $\Omega$ of $\mathbb{R}^{n}(n\geq2)$. Firstly, we show the nondegeneracy 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 curved surfaces at the nondegenerate critical points. Secondly, we deduce the uniqueness and nondegeneracy of the critical points of solutions in a rotationally symmetric domain of $\mathbb{R}^{n}(n\geq3)$ by the projection of higher dimension space onto two dimension plane.