On the optimal shape of a thin insulating layer

We are interested in the thermal insulation of a bounded open set $\mathrm{\Omega }$ surrounded by a set whose thickness is locally described by $\epsilon h$, where $h$ is a non-negative function defined on the boundary $\mathrm{\partial }\mathrm{\Omega }$. We study the problem in the limit for $\epsilon$ going to zero using a first-order asymptotic development by $\mathrm{\Gamma }$-convergence.