Mathematics

We study the asymptotic behaviour of solutions to variational problems in perforated
domains with Neumann boundary conditions. We consider perforations that in the limit
concentrate on a smooth manifold. We characterise the limits of the solutions and show that they solve a variational problem with a transmission condition across the manifold. This is expressed through a measure on the manifold, vanishing on sets of capacity zero. Then, we prove that every such measure can be obtained by homogenising suitable perforations. Eventually, we provide an asymptotic formula for this measure by using some auxiliary minimum problems.