Transmission conditions obtained by homogenisation

Given a bounded open set in Rn, n≥2, and a sequence (Kj) of compact sets converging to an (n-1)-dimensional manifold M, we study the asymptotic behaviour of the solutions to some minimum problems for integral functionals on Ω\Kj, with Neumann boundary conditions on ∂(Ω\Kj). We prove that the limit of these solutions is a minimiser of the same functional on Ω\M subjected to a transmission condition on M, which can be expressed through a measure µ supported on M. The class of all measures that can be obtained in this way is characterised, and the link between the measure µ and the sequence (Kj) is expressed by means of suitable local minimum problems.