Periodic homogenization of Dirichlet problem for divergence type elliptic operators

Abstract

The thesis studies homogenization of Dirichlet boundary value problems for divergence type elliptic operators, and the associated boundary layer issues. This type of problems for operators with periodically oscillating coeffcients, and fixed boundary data are by now a classical topic largely due to the celebrated work by Avellaneda and Lin from late 80's. The case when the operator and the Dirichlet boundary data exhibit periodic oscillations simultaneously was a longstanding open problem, and a progress in this direction has been achieved only very recently, in 2012, by Gerard-Varet and Masmoudi who proved a homogenization result for the simultaneously oscillating case with an algebraic rate of convergence in L2. Aimed at understanding the homogenization process of oscillating boundary data, in the first part of the thesis we introduce and develop Fourier-analytic ideas into the study of homogenization of Dirichlet boundary value problems for elliptic operators in divergence form. In smooth and bounded domains, for fixed operator and periodically oscillating boundary data we prove pointwise, as well as Lp convergence results the homogenization problem. We then investigate the optimality (sharpness) of our Lp upper bounds. Next, for the above mentioned simultaneously oscillating problem studied by Gerard-Varet and Masmoudi, we establish optimal Lp bounds for homogenization in some class of operators. For domains with non smooth boundary, we study similar boundary value homogenization problems for scalar equations set in convex polygonal domains. In the vein of smooth boundaries, here as well for problems with fixed operator and oscillating Dirichlet data we prove pointwise, and Lp convergence results, and study the optimality of our Lp bounds. Although the statements are somewhat similar with the smooth setting, challenges for this case are completely different due to a radical change in the geometry of the domain. The second part of the work is concerned with the analysis of boundary layers arising in periodic homogenization. A key difficulty toward the homogenization of Dirichlet problem for elliptic systems in divergence form with periodically oscillating coefficients and boundary condition lies in identification of the limiting Dirichlet data corresponding to the effective problem. This question has been addressed in the aforementioned work by Gerard-Varet and Masmoudi on the way of proving their main homogenization result. Despite the progress in this direction, some very basic questions remain unanswered, for instance the regularity of this effective data on the boundary. This issue is directly linked with the up to the boundary regularity of homogenized solutions, but perhaps more importantly has a potential to cast light on the homogenization process. We initiate the study of this regularity problem, and prove certain Lipschitz continuity result. The work also comprises a study on asymptotic behaviour of solutions to boundary layer systems set in halfspaces. By a new construction we show that depending on the normal direction of the hyperplane, convergence of the solutions toward their tails far away from the boundaries can be arbitrarily slow. This last result, combined with the previous studies gives an almost complete picture of the situation.