Semilinear elliptic equations in thin regions with terms concentrating on oscillatory boundaries

Abstract

In this work we study the behavior of a family of solutions of a semilinear elliptic equation, with homogeneous Neumann boundary condition, posed in a two-dimensional oscillating thin region with reaction terms concentrated in a neighborhood of the oscillatory boundary. Our main result is concerned with the upper and lower semicontinuity of the set of solutions. We show that the solutions of our perturbed equation can be approximated with one of a one-dimensional equation, which also captures the effects of all relevant physical processes that take place in the original problem.