A quantitative symmetry result for $p$-Laplace equations with discontinuous nonlinearities

TL;DR

This paper establishes a quantitative stability estimate for symmetry of solutions to the $p$-Laplace equation with discontinuous nonlinearities, linking deviation from symmetry to the domain's isoperimetric deficit.

Contribution

It provides a quantitative version of the Gidas-Ni-Nirenberg symmetry result for $p$-Laplace equations with discontinuous nonlinearities, extending previous qualitative results.

Findings

Deviation of solutions from Schwarz symmetrization is bounded by the isoperimetric deficit.

The approach uses a quantitative Pólya-Szegő principle.

Results apply to solutions in bounded domains with discontinuous nonlinearities.

Abstract

In this paper, we study positive solutions $u$ of the homogeneous Dirichlet problem for the $p$-Laplace equation $-\Delta_p \,u=f(u)$ in a bounded domain $\Omega\subset\mathbb{R}^N$, where $N\ge 2$, $1<p<+\infty$ and $f$ is a discontinuous function. We address the quantitative stability of a Gidas-Ni-Nirenberg type symmetry result for $u$, which was established by Lions and Serra when $\Omega$ is a ball. By exploiting a quantitative version of the P\'olya-Szeg\"o principle, we prove that the deviation of $u$ from its Schwarz symmetrization can be estimated in terms of the isoperimetric deficit of $\Omega$.