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 $$\{x_3=0\}$$ { x 3 = 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 $$\varepsilon $$ ε , size of the small regions $$r_\varepsilon $$ r ε and Robin parameter $$\beta (\varepsilon )$$ β ( ε ) . In particular, we address the convergence, as $$\varepsilon $$ ε tends to zero, of the solutions for the critical size of the small regions $$r_\varepsilon =O(\varepsilon ^{ 2})$$ r ε = O ( ε 2 ) . For certain $$\beta (\varepsilon )$$ 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.