Isoperimetric sets for weighted twisted eigenvalues

TL;DR

This paper establishes an isoperimetric inequality for the first twisted eigenvalue of a weighted operator, identifying the sets that minimize this eigenvalue under a weighted measure, with implications for understanding optimal shapes in weighted spaces.

Contribution

It introduces a new isoperimetric inequality for weighted twisted eigenvalues and characterizes the optimal sets minimizing these eigenvalues.

Findings

Optimal sets are two disjoint copies of classical isoperimetric sets.

The inequality applies to measures with positive density functions.

Identifies shapes minimizing the first twisted eigenvalue in weighted spaces.

Abstract

In tis paper we prove an isoperimetric inequality for the first twisted eigenvalue $\lambda_{1,\gamma}^T(\Omega)$ of a weighted operator, defined as the minimum of the usual Rayleigh quotient when the trial functions belong to the weighted Sobolev space $H_0^1(\Omega,d\gamma)$ and have weighted mean value equal to zero in $\Omega$. We are interested in positive measures $d\gamma=\gamma(x) dx$ for which we are able to identify the isoperimetric sets, namely, the sets that minimize $\lambda_{1,\gamma}^T(\Omega)$ among sets of given weighted measure. In the cases under consideration, the optimal sets are given by two identical and disjoint copies of the isoperimetric sets (for the weighted perimeter with respect to the weighted measure).