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 $u$.
Categories: math.AP