Extremal Eigenvalue Problems for Starlike Planar Domains

Each eigenvalue of the Laplacian, subject to Dirichlet boundary conditions, is shown to attain its extremes over those open planar starlike sets that simultaneously (i) contain a given disk, (ii) occupy a given area, and (iii) do not exceed a prescribed perimeter. Over that subclass of starlike sets with Lipschitz boundary we compute the generalized gradient, with respect to domain, of each eigenvalue and deduce, from the ensuing necessary conditions, partial regularity of the extremal domains.