Regularity for Shape Optimizers: The Degenerate Case

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.