Regularity for Shape Optimizers: The Nondegenerate Case

Abstract

We consider minimizers of \[ F(λ_1(Ω),\ldots,λ_N(Ω)) + |Ω|, \] where $F$ is a function strictly increasing in each parameter, and $\hat{I}{»}_k(\hat{I}\text{c◯})$ is the $k$-th Dirichlet eigenvalue of $\hat{I}\text{c◯}$. Our main result is that the reduced boundary of the minimizer is composed of $C^{1,\hat{I}\pm }$ 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.

minor fixes