Boundary homogenization with large reaction terms on a strainer-type wall

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 { x3= 0 }. On this part, the boundary conditions alternate from Neumann to nonlinear-Robin, being of Dirichlet type outside. The nonlinear-Robin boundary conditions are imposed on small regions periodically placed along the plane and contain a Robin parameter that can be very large. We provide all the possible homogenized problems, depending on the relations between the three parameters: period ε, size of the small regions rε and Robin parameter β(ε). In particular, we address the convergence, as ε tends to zero, of the solutions for the critical size of the small regions rε= O(ε2). For certain β(ε) , a nonlinear capacity term arises in the strange term which depends on the macroscopic variable and allows us to extend the usual capacity definition to semilinear boundary conditions.