A quantitative version of the Gidas-Ni-Nirenberg Theorem

TL;DR

This paper investigates the stability of the Gidas-Ni-Nirenberg symmetry result for positive solutions of semilinear equations under small perturbations, providing a quantitative analysis of how solutions deviate from radial symmetry.

Contribution

It introduces a quantitative stability version of the classical symmetry theorem for semilinear equations, extending the understanding of solution behavior under perturbations.

Findings

Established a stability estimate quantifying deviation from radial symmetry

Proved that solutions remain close to symmetric solutions under small perturbations

Extended classical symmetry results to a quantitative framework

Abstract

A celebrated result by Gidas, Ni & Nirenberg asserts that classical positive solutions to semilinear equations $- \Delta u = f(u)$ in a ball vanishing at the boundary must be radial and radially decreasing. In this paper we consider small perturbations of this equation and study the quantitative stability counterpart of this result.