Flux terms and Robin boundary conditions as limit of reactions and potentials concentrating at the boundary

Abstract

We analyze the limit of the solutions of an elliptic problem when some reaction and potential terms are concentrated in a neighborhood of a portion $\Gamma$ of the boundary and this neighborhood shrinks to $\Gamma$ as a parameter goes to zero. We prove that this family of solutions converges in certain Sobolev spaces and also in the sup norm, to the solution of an elliptic problem where the reaction term and the concentrating potential are transformed into a flux condition and a potential on $\Gamma$.