Homogenization for Alternating Boundary Conditions with Large Reaction Terms Concentrated in Small Regions

Abstract

We consider a homogenization problem for the Laplace operator posed in a bounded domain of the upper half-space, a part of its boundary being in contact with the plane. On this part, the boundary conditions alternate from Neumann to Robin, being of Dirichlet type outside. The Robin conditions are imposed on small regions periodically placed along the plane and contain a Robin parameter that can be very large. The period tends to zero, and we provide all the possible homogenized problems, depending on the relations between the three parameters: period, size of the small regions, and Robin parameter. We address the convergence of the solutions in the most critical case where a non-constant capacity coefficient arises in the strange term.