Renormalized solutions to a class of quasilinear elliptic problems with weak data : existence, uniqueness and homogenization

Overview & Synopsis

In this thesis, we study a class of quasilinear elliptic equations posed in atwo-component domain with an L1 data and its asymptotic analysis. More precisely, we consider a two-component domain, denoted by Ω, which can be written as the disjoint union Ω = Ω 1 ∪ Ω 2 ∪ Г, where the open sets Ω 1 and Ω 2 are the two components of Ω, and Г is the interface between thesecomponents. We study the following quasilinear elliptic problem posed in Ω:−div(B(x, u1)∇u1) = f in Ω1,−div(B(x, u2)∇u2) = f in Ω2,(B(x, u1)∇u1)υ1 = (B(x, u2)∇u2)υ1 on Г,(B(x, u1)∇u1)υ1 = −h(x)(u1 − u2) on Г,u1 = 0 on ∂Ω,where υ1 is the unit outward normal to Ω1, f is an L1 function, and B is a coercive matrix field which has a restricted growth assumption (B(x, r) is bounded on any compact set of R). The first part of this thesis is dedicated to existence and uniqueness results for this problem in the framework of renormalized solutions, which was introduced by R.J. DiPerna and P.L. Lions. In the second part, 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. Damlamian and G. Griso. It has been successively adapted to two-component domains by P. Donato, K.H. Le Nguyen, and R. Tardieu. In order to obtain a uniqueness result for the homogenized problem, we study the properties of the corresponding cell problem. In particular, we show that if the matrix field in the cell problem, denoted A(y, t), is local Lipschitzcontinuous with respect to t, then the resulting homogenizedmatrix A0 keeps this property. 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).