An isoperimetric inequality for a nonlinear eigenvalue problem

TL;DR

This paper establishes an isoperimetric inequality for a nonlinear eigenvalue problem, demonstrating that the optimal shape minimizing the eigenvalue for a given volume is the union of two equal balls.

Contribution

It extends classical isoperimetric inequalities to a nonlinear eigenvalue context, identifying the union of two equal balls as the minimizer.

Findings

The minimizer for the nonlinear eigenvalue problem is the union of two equal balls.

The inequality generalizes the Rayleigh-Faber-Krahn inequality to nonlinear settings.

The result characterizes the optimal shape among sets of fixed volume.

Abstract

We prove an isoperimetric inequality of the Rayleigh-Faber-Krahn type for a nonlinear generalization of the first twisted Dirichlet eigenvalue. More precisely, we show that the minimizer among sets of given volume is the union of two equal balls.