A spectral shape optimization problem with a nonlocal competing term

TL;DR

This paper investigates a spectral shape optimization problem involving the first eigenvalue of the Dirichlet Laplacian combined with a Riesz-type interaction, identifying conditions for existence, optimality, and non-existence of minimizers.

Contribution

It introduces a new spectral optimization model with a nonlocal Riesz term and characterizes the existence and shape of minimizers depending on the Riesz interaction strength.

Findings

Existence of minimizers when Riesz strength is below a critical value.

The ball is a rigid minimizer for small Riesz repulsion.

Regular minimizers do not exist in certain Riesz regimes.

Abstract

We study the minimization of a spectral functional made as the sum of the first eigenvalue of the Dirichlet Laplacian and the relative strength of a Riesz-type interaction functional. We show that when the Riesz repulsion strength is below a critical value, existence of minimizers occurs. Then we prove, by means of an expansion analysis, that the ball is a rigid minimizer when the Riesz repulsion is small enough. Eventually we show that for certain regimes of the Riesz repulsion, regular minimizers do not exist.