Regularity for Shape Optimizers: The Degenerate Case

TL;DR

This paper proves that minimizers of a class of shape optimization problems involving eigenvalues have smooth boundaries, using an approximation method to derive necessary optimality conditions despite degeneracies.

Contribution

It introduces a vanishing viscosity approach to establish boundary regularity and derive Euler-Lagrange equations for degenerate shape optimization problems.

Findings

Minimizers have smooth boundary graphs.

Reduced boundary is composed of smooth surfaces.

Method handles degeneracies in eigenvalue-based optimization.

Abstract

We consider minimizers of \[ F(\lambda_1(\Omega),\ldots,\lambda_N(\Omega)) + |\Omega|, \] where $F$ is a function nondecreasing in each parameter, and $\lambda_k(\Omega)$ is the $k$-th Dirichlet eigenvalue of $\Omega$. This includes, in particular, functions $F$ which depend on just some of the first $N$ eigenvalues, such as the often studied $F=\lambda_N$. The existence of a minimizer, which is also a bounded set of finite perimeter, was shown recently. Here we show that the reduced boundary of the minimizers $\Omega$ is made up of smooth graphs, and examine the difficulties in classifying the singular points. Our approach is based on an approximation ("vanishing viscosity") argument, which--counterintuitively--allows us to recover an Euler-Lagrange equation for the minimizers which is not otherwise available.