HOMOGENIZATION OF ROBIN AND STEKLOV PROBLEMS IN A DOMAIN WITH A THIN ROUGH BOUNDARY

In this paper, we study the asymptotic behavior of some elliptic problems posed in a domain with a rough periodically oscillating boundary, where the height of the oscillations goes to zero. First, we prove the ho- mogenization and corrector results for the problem with a Robin boundary condition, obtaining different limit problems, according to the height of the oscillations and on the boundary condition. Then, we apply these results to the related spectral problem, proving the convergence of the eigenvalues and the corresponding eigenspaces to those of the homogenized problems. In the second part, we show similar results for the associated Steklov prob- lem. To this aim, we introduce a suitable problem, whose homogenization is also investigated.