Nonlinear elliptic equations with concentrating reaction terms at an oscillatory boundary

Abstract

<p style='text-indent:20px;'>In this paper we analyze the asymptotic behavior of a family of solutions of a semilinear elliptic equation, with homogeneous Neumann boundary condition, posed in a two-dimensional oscillating region with reaction terms concentrated in a neighborhood of the oscillatory boundary $\theta_\varepsilon \subset\Omega_{\varepsilon }\subset \mathbb{R}^2$ when a small parameter $\varepsilon &gt;0$ goes to zero. Our main result is concerned with the upper and lower semicontinuity of the set of solutions in $H^1$. We show that the solutions of our perturbed equation can be approximated with one defined in a fixed limit domain, which also captures the effects of reaction terms that take place in the original problem as a flux condition on the boundary of the limit domain.