Homogeneização para uma classe de funcionais integrais

Abstract

The main subject of this thesis is the study of the homogenization of stochastic integral functionals of the form Fε(ω)(u, D) = ∫ D f(ω, Φ −1 ( x ε , ω),∇u)dx, where the integrand f is an admissible integrand and Φ is a stochastic deformation. Our main result shows the Γ-convergence of the family of random integral functionals (Fε) as ε → 0 to an integral functional F0 of the form F0(u, D) = ∫ D f0(∇u(x))dx. In this thesis, we generalize the stochastic results obtained by Dal Maso-Modica in [16], [17], [18], based on the Subadditive Ergodic Theorem of Akcoglu-Krengel [1], by considering a class of functionals with stationary coefficients composed with a stochastic deformation. The proof of our main homogenization result is based on the generalization of an ergodic theorem for the composition of a subadditive process with a stochastic deformation and Γ-convergence theorems of our functionals.