Extreme Cases in Boundary Homogenization for the Linear Elasticity System

We consider a homogenization problem for the linear elasticity system posed in a domain \(\varOmega \) of the upper half-space, a part of its boundary \(\varSigma \) being in contact with the plane. We assume that the surface \(\varSigma \) is traction-free out of small regions \(T^\varepsilon \) , where we impose Winkler-Robin boundary conditions. The regions are at a distance \(O(\varepsilon )\) between them. This condition links stresses and displacements by means of a symmetric and positive definite matrix-function and a reaction parameter that can be very large when \(\varepsilon \to 0\) . We address the convergence, as \(\varepsilon \to 0\) , of the solutions in the extreme cases where the averaged boundary condition on the plane is a Dirichlet or a Neumann one. A certain non-periodical distribution of the reaction regions is allowed. We also address the convergence of the associated spectral problems.