Regularity for Shape Optimizers: The Degenerate Case

We consider minimizers of F=(λ1(Ω),..λN(Ω))+ |Ω| where F is a function nondecreasing in each parameter, and λ_k(Ω) is the k^th Dirichlet eigenvalue of ω. This includes, in particular, functions F that depend on just some of the first N eigenvalues, such as the often-studied F=λ_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 Ω 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 that is not otherwise available.