Shape optimization of a thermal insulation problem

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