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 $\widetilde{\Sigma}_{\lambda}^{\nu}=\{x\in\Omega:x^{\ast}\in\Sigma_{\lambda}^{\nu}\}$ 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)$\,\frac{\partial u}{\partial\nu}(x)>0\quad\forall x\in \Sigma_{\lambda_{\nu}}^{\nu}$ and (ii)$\,\frac{\partial u}{\partial\nu}(x)<0\quad\forall x\in \widetilde{\Sigma}_{\lambda_{\nu}}^{\nu}$. 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 $u$.