# The Soap Bubble Theorem and a $p$-Laplacian overdetermined problem

## TL;DR

This paper proves that for a specific $p$-Laplacian problem, constant boundary mean curvature implies the domain is spherical, extending classical symmetry results to a nonlinear PDE setting.

## Contribution

It establishes a spherical symmetry result for the $p$-Laplacian overdetermined problem with constant mean curvature boundary, generalizing classical theorems.

## Key findings

- Domains with constant boundary mean curvature are spheres.
- Unique solution is radially symmetric.
- Uses integral identities, $P$-function, and maximum principle.

## Abstract

We consider the $p$-Laplacian equation $-\Delta_p u=1$ for $1<p<2$, on a regular bounded domain $\Omega\subset\mathbb R^N$, with $N\ge2$, under homogeneous Dirichlet boundary conditions. In the spirit of Alexandrov's Soap Bubble Theorem and of Serrin's symmetry result for the overdetermined problems, we prove that if the mean curvature $H$ of $\partial\Omega$ is constant, then $\Omega$ is a ball and the unique solution of the Dirichlet $p$-Laplacian problem is radial. The main tools used are integral identities, the $P$-function, and the maximum principle.