Regularity for Shape Optimizers: The Nondegenerate Case

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.