On the optimal shape of a thin insulating layer (Q6561294)

The authors deal with the problem of the thermal insulation of a bounded open set \(\Omega\) surrounded by a set whose thickness is locally described by \(\varepsilon h,\ h\ge 0,\ h:\partial\Omega\to \mathbb{R}.\) They consider the minimization of the energy functional \N\[\N\mathcal{F}_\varepsilon(v,h)= \varepsilon\int_{\Sigma_\varepsilon}|\nabla v|^2\,dx+ \beta\int_{\partial\Omega_\varepsilon}v^2\,d\mathcal{H}^{n-1},\N\]\Nwhere \(v\in H^1(\Omega_\varepsilon)\) with \(v=1\) in \(\Omega\) and \N\[\N\Sigma_\varepsilon=\{\sigma + t\nu(\sigma)|\sigma \in \partial\Omega,\ 0<t<\varepsilon h(\sigma)\}, \ \Omega_\varepsilon=\bar\Omega\cup \Sigma_\varepsilon.\N\]\N\(\Gamma-\)convergence is the main tool for a study of the problem as \(\varepsilon\) goes to zero with the use off the first-order asymptotic development. It is the main result of the paper.