Homogenization technique in inverse problems for boundary hemivariational inequalities

The purpose of the article is to present a methodology which is useful in the derivation of approximate models with simpler geometry of some inverse problems. First we formulate a direct problem (being the boundary hemivariational inequality) which is given in a domain with a complicated geometry (e.g. perforated domains, layered structures). For such direct problem we consider the inverse one and we provide result on the existence of solutions. Next we establish the homogenization result for the direct problem. It turns out that in the homogenized inequality the complex boundary condition is replaced by a much simpler one. Finally, we study the asymptotic behavior of the set of solutions of the inverse problem. The main result shows that the solutions to the inverse problem for homogenized hemivariational inequality can be considered as reasonable approximations of the solutions of the original inverse problem.