Patrizia Donato, Renormalized Solution-Quasilinear PDE (Jump)-Existence, Uniqueness, Homogenization

We consider a class of quasilinear elliptic equations posed in a two-component domain with an L^1data with a coercive matrix field B(x, u) which has a restricted growth assumption, that is, it is only bounded on any compact set of R. We first present existence and uniqueness results for this problem in the framework of renormalized solutions, introduced by R. J. DiPerna and P. L. Lions. Then, we study the corresponding homogenization problem for a two-component domain with a (disconnected) periodic second component, by combining the notion of renormalized solutions and the periodic unfolding method, introduced D. Cioranescu, A. Damlamianand G. Griso. In order to obtain a uniqueness result for the homogenized problem, we give a comparison result for the solutions of the cell problem. This uniqueness result ensures that the convergences obtained in the homogenization process hold for the whole sequence of the periodicity parameter (and not only a subsequence).