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

Haiyun Deng; Hairong Liu; Long Tian

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arXiv · 1712.08454

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

Haiyun Deng·Hairong Liu·Long Tian

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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.

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Haiyun Deng, Hairong Liu, Long Tian. 2017-12-22. Uniqueness of critical points of solutions to the mean curvature equation with Neumann and Robin boundary conditions. https://arxiv.org/abs/1712.08454

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