Dimension reduction and homogenization of random degenerate operators. Part I
Abdelaziz Aït Moussa, Loubna Zlaïji · LMS Journal of Computation and Mathematics · 2012
Abstract Our aim in this paper is to identify the limit behavior of the solutions of random degenerate equations of the form −div Aε(x′,∇Uε)+ρεω(x′)Uε=Fwith mixed boundary conditions on Ωεwhenever ε→0, where Ωεis anN-dimensional thin domain with a small thicknessh(ε),ρεω(x′)=ρω(x′/ε), whereρωis the realization of a random functionρ(ω) , andAε(x′,ξ)=a(Tx′/εω,ξ) , the mapa(ω,ξ) being measurable inωand satisfying degenerated structure conditions with weightρinξ. As usual in dimension reduction problems, we focus on the rescaled equations and we prove that under the conditionh(ε)/ε→0 , the sequence of solutions of them converges to a limitu0, whereu0is the solution of an (N−1) -dimensional limit problem with homogenized and auxiliary equations.