Shape optimization of a thermal insulation problem

We study a shape optimization problem involving a solid K⊂ℝ^n that is maintained at constant temperature and is enveloped by a layer of insulating material Ω which obeys a generalized boundary heat transfer law. We minimize the energy of such configurations among all (K,Ω ) with prescribed measure for K and Ω , and no topological or geometrical constraints. In the convection case (corresponding to Robin boundary conditions on ∂Ω ) 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.