Spectral Boundary Homogenization Problems in Perforated Domains with Robin Boundary Conditions and Large Parameters

In this chapter we address asymptotics for spectral problems posed in periodically perforated domains along a plane. The operator under consideration is the Laplacian, and the spectral problem is posed in a three-dimensional domain Ω, outside the cavities. The boundary conditions are of the Dirichlet type on the boundary of Ω and of the Robin type on the boundary of the cavities. The periodicity of the structure is?; it is a small parameter that converges towards zero. The size of the cavities can be of the same order of magnitude as?, namely O(?), or much smaller than?, namely o(?). Also a large?-dependent parameter (adsorption constant) arises in the Robin conditions. Depending on the different values/relations between the three parameters (periodicity, size of cavities and adsorption constant) different homogenized problems are obtained: both critical sizes for cavities and critical relations for parameters are provided. The results complement earlier ones, where the convergence for the spectrum is outlined when dealing with linear problems. Here, we obtain estimates for convergence rates of the eigenvalues and eigenfunctions in terms of the eigenvalue number and the parameter?.