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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 $\Omega$. Thisincludes, 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 aminimizer, 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 ofsmooth 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 forthe minimizers which is not otherwise available.

Regularity for Shape Optimizers: The Degenerate Case