# Optimization results for the higher eigenvalues of the $p$-Laplacian associated with sign-changing capacitary measures

## TL;DR

This paper establishes the existence of optimal sets for the $p$-Laplacian eigenvalues and Schrödinger eigenvalues within fixed measure constraints, using a novel approach to handle nonlinear shape optimization problems involving sign-changing measures.

## Contribution

It introduces a general method to analyze the continuous dependence of eigenvalues on sign-changing capacitary measures under $oldsymbol{ extgamma}$-convergence, enabling existence proofs for optimal sets.

## Key findings

- Existence of optimal sets for the $k$-th eigenvalue of the $p$-Laplacian.
- Existence of optimal potentials for Schrödinger eigenvalues.
- Development of a new approach for nonlinear shape optimization with sign-changing measures.

## Abstract

In this paper we prove the existence of an optimal set for the minimization of the $k$-th variational eigenvalue of the $p$-Laplacian among $p$-quasi open sets of fixed measure included in a box of finite measure. An analogous existence result is obtained for eigenvalues of the $p$-Laplacian associated with Schr\"odinger potentials. In order to deal with these nonlinear shape optimization problems, we develop a general approach which allows to treat the continuous dependence of the eigenvalues of the $p$-Laplacian associated with sign-changing capacitary measures under $\gamma$-convergence.