  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.
