Regularity for Shape Optimizers: The Degenerate Case

Abstract

We consider minimizers of \[ F(λ_1(Ω),\ldots,λ_N(Ω)) + |Ω|, \] where $F$ is a function nondecreasing in each parameter, and $\hat{I}{»}_k(\hat{I}\text{c◯})$ is the $k$-th Dirichlet eigenvalue of $\hat{I}\text{c◯}$. This includes, in particular, functions $F$ which depend on just some of the first $N$ eigenvalues, such as the often studied $F=\hat{I}{»}_N$. 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 $\hat{I}\text{c◯}$ 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.

Minor typos fixed