A note on the radially symmetry in the moving plane method

TL;DR

This paper proves that under certain symmetry and monotonicity conditions, a function and domain are radially symmetric, extending classical results and providing a simpler proof for symmetry in bounded domains and the whole space.

Contribution

It offers a simplified proof that functions satisfying specific symmetry and monotonicity conditions are radially symmetric, extending classical symmetry results to broader contexts.

Findings

Functions with symmetry and monotonicity conditions are radially symmetric.

Domain symmetry about a plane implies overall radial symmetry.

Results extend to the entire space with weaker assumptions.

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 $u$.