Regularity for Shape Optimizers: The Nondegenerate Case

TL;DR

This paper proves regularity properties of shape optimizers minimizing a function of Dirichlet eigenvalues plus volume, showing their boundaries are mostly smooth and establishing new results for related free boundary problems.

Contribution

It establishes regularity of minimizers' boundaries in shape optimization problems involving eigenvalues, and introduces new regularity results for vector-valued Bernoulli free boundary problems.

Findings

Reduced boundary consists of $C^{1,eta}$ graphs.

Topological boundary is mostly regular, with singular set of Hausdorff dimension at most $n-3$.

New regularity results for vector-valued Bernoulli free boundary problems.

Abstract

We consider minimizers of \[ F(\lambda_1(\Omega),\ldots,\lambda_N(\Omega)) + |\Omega|, \] where $F$ is a function strictly increasing in each parameter, and $\lambda_k(\Omega)$ is the $k$-th Dirichlet eigenvalue of $\Omega$. Our main result is that the reduced boundary of the minimizer is composed of $C^{1,\alpha}$ graphs, and exhausts the topological boundary except for a set of Hausdorff dimension at most $n-3$. We also obtain a new regularity result for vector-valued Bernoulli type free boundary problems.