Shape optimization of a thermal insulation problem

Abstract

We study a shape optimization problem involving a solid Ksubset {mathbb {R}}^n that is maintained at constant temperature and is enveloped by a layer of insulating material Omega which obeys a generalized boundary heat transfer law. We minimize the energy of such configurations among all (K,Omega ) with prescribed measure for K and Omega , and no topological or geometrical constraints. In the convection case (corresponding to Robin boundary conditions on partial Omega ) we obtain a full description of minimizers, while for general heat transfer conditions, we prove the existence and regularity of solutions and give a partial description of minimizers.