Regularity result for a shape optimization problem under perimeter constraint

TL;DR

This paper investigates the shape optimization of Dirichlet Laplace eigenvalues under perimeter constraints, establishing regularity and eigenvalue gap results for optimal shapes.

Contribution

It proves analyticity of optimal sets outside a small singular set and shows the strict ordering of optimal eigenvalues, advancing understanding of shape optimization under perimeter constraints.

Findings

Optimal sets are analytic outside a small singular set.

The k-th eigenvalue is strictly less than the (k+1)-th eigenvalue for optimal shapes.

Provides elliptic regularity results for sets with bounded weak curvature.

Abstract

We study the problem of optimizing the eigenvalues of the Dirichlet Laplace operator under perimeter constraint. We prove that optimal sets are analytic outside a closed singular set of dimension at most $d-8$ by writing a general optimality condition in the case the optimal eigenvalue is multiple. As a consequence we find that the optimal $k$-th eigenvalue is strictly smaller than the optimal $(k+1)$-th eigenvalue. We also provide an elliptic regularity result for sets with positive and bounded weak curvature.