Stability Results for the Robin-Laplacian on Nonsmooth Domains

We formulate a generalization of the Laplace equation under Robin boundary conditions on a large class of possibly nonsmooth domains by dealing with the trace term appearing in the variational formulation from the point of view of the theory of functions of bounded variation. Admissible domains may have inner boundaries, i.e., inner cracks. In dimension two, we formulate a stability result for the elliptic problems under domain variation: with this aim, we introduce a notion of perimeter (Robin perimeter) which is tailored to count the inner boundaries with the appropriate natural multiplicity.