Shape Optimization Problems with Internal Constraint

Abstract

We consider shape optimization problems with internal inclusion constraints, of the form \min\big\{J(\Omega)\ :\ \Dr\subset\Omega\subset\R^d,\ |\Omega|=m\big\}, where the set \Dr is fixed, possibly unbounded, and $J$ depends on $\Omega$ via the spectrum of the Dirichlet Laplacian. We analyze the existence of a solution and its qualitative properties, and rise some open questions.Comment: 18 pages, 0 figure