Stochastic Diffeomorphisms and Homogenization of Multiple Integrals
Antoine Gloria · Applied Mathematics Research eXpress · 2008
In [4], Blanc, Le Bris, and Lions have introduced the notion of stochastic diffeomorphism together with a variant of stochastic homogenization theory for linear and monotone elliptic operators.Their proofs rely on the ergodic theorem and on the analysis of the associated corrector equation.In the present article, we provide another proof of their results using the formalism of integral functionals.We also extend the analysis to cover the case of quasiconvex integrands. IntroductionIn [4], Blanc, Le Bris, and Lions have introduced the notion of stochastic diffeomorphism together with a variant of stochastic homogenization theory for linear and monotone elliptic operators.Their proofs rely on the ergodic theorem and on the analysis of the associated corrector equation in an abstract probability space, combined with Tartar's oscillating test function method (or two-scale convergence).In [7], they draw the link between this stochastic variant of the homogenization theory and their previous work on stochastic lattices in [5,6].Using another classical approach to homogenization theory, we give an alternative proof of (some of) their results, and extend them to the quasiconvex case.Our proof, which closely follows the one by Dal Maso and Modica in [12], is based on the compactness of a class of integral functionals with respect to -convergence,