On some homogenization problems in perforated domains with nonlinear boundary conditions

Abstract

In a perforated domain with ε-periodic structure we consider a linear second order elliptic equation with repidly oscillating ε coefficients, where εis a small positive parameter. Zero Dirichlet conditions are prescribed on the outer portion of the boundary.On the surface of the perforation the conormal derivative is expressed through the unknown function by a nonlinear relation involving the small paremeter ε According to the typeof this relation and its dependence on ε various types of homogenized problems are obtained,together with estimates which charaterize the convergence of solutions and their gradients as In particular, we consider some nonlioner boundary conditions on the perforation which result in the homogenized problem that can only be expressed in terms of a variational inequelity. As an application,we establish a homogenization theorem for a problem of Signorini type in a perforated domain.