A note on the radially symmetry in the moving plane method

Abstract

Let $\Omega \subset {\mathbb{R}}^n$, $n\ge 2$, be a bounded connected $C^2$ domain. For any unit vector $\nu \in {\mathbb{R}}^n$, let $T_{\lambda }^{\nu }=\{x\in {\mathbb{R}}^n:x\cdot \nu =\lambda \}$, ${\Sigma }_{\lambda }^{\nu }=\{x\in \Omega :x\cdot \nu <\lambda \}$ and $x^{\ast }=x-2(x\cdot \nu -\lambda )\nu$ be the reflection of a point $x\in {\mathbb{R}}^n$ about the plane $T_{\lambda }^{\nu }$. Let ${\tilde{\Sigma }}_{\lambda }^{\nu }=\{x\in \Omega :x^{\ast }\in {\Sigma }_{\lambda }^{\nu }\}$ and $u\in C^2(\overline{\Omega })$. Suppose for any unit vector $\nu \in {\mathbb{R}}^n$, there exists a constant ${\lambda }_{\nu }\in \mathbb{R}$ such that $\Omega$ is symmetric about the plane $T_{{\lambda }_{\nu }}^{\nu }$ and $u$ is symmetric about the plane $T_{{\lambda }_{\nu }}^{\nu }$ and satisfies (i)$\frac{\partial u}{\partial \nu }(x)>0\forall x\in {\Sigma }_{{\lambda }_{\nu }}^{\nu }$ and (ii)$\frac{\partial u}{\partial \nu }(x)<0\forall x\in {\tilde{\Sigma }}_{{\lambda }_{\nu }}^{\nu }$. We will give a simple proof that $u$ is radially symmetric about some point $x_0\in \Omega$ and $\Omega$ is a ball with center at $x_0$. Similar result holds for the domain ${\mathbb{R}}^n$ and function $u\in C^2({\mathbb{R}}^n)$ satisfying similar monotonicity and symmetry conditions. We also extend this result under weaker hypothesis on the function $u$.

6 pages