Rigidity results for the $p$-Laplacian Poisson problem with Robin boundary conditions

TL;DR

This paper investigates the conditions under which equality holds in a Talenti-type comparison for the $p$-Laplacian Poisson problem with Robin boundary conditions, establishing that equality implies the domain is a ball and functions are radial.

Contribution

It proves that equality in the Talenti comparison for the $p$-Laplacian with Robin conditions occurs only for radial functions on spherical domains, extending rigidity results.

Findings

Equality in the comparison implies the domain is a ball.

Both the solution and the right-hand side must be radial.

The result characterizes the symmetry of solutions under equality conditions.

Abstract

Let $\Omega \subset \mathbb{R}^n$ be an open, bounded and Lipschitz set. We consider the Poisson problem for the $p-$Laplace operator associated to $\Omega$ with Robin boundary conditions. In this setting, we study the equality case in the Talenti-type comparison stated in \cite{AGM}. We prove that the equality is achieved only if $\Omega$ is a ball and both the function $u$ and the right hand side $f$ of the Poisson equation are radial.