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Dimension reduction and homogenization of random degenerate operators. Part I

Part of: Stochastic processes Partial differential equations Miscellaneous topics - Partial differential equations Elliptic equations and systems Stochastic analysis

Published online by Cambridge University Press:  01 January 2012

Abdelaziz Aït Moussa  and
Loubna Zlaïji
Abdelaziz Aït Moussa
Affiliation:
Department of Mathematics and Computer Sciences, University of Mohammed Premier, 60040 Oujda, Morocco
Loubna Zlaïji
Affiliation:
Department of Mathematics and Computer Sciences, University of Mohammed Premier, 60040 Oujda, Morocco (email: l.zlaiji@yahoo.fr)

Abstract

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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=F with mixed boundary conditions on Ωε whenever ε→0, where Ωε is an N-dimensional thin domain with a small thickness h(ε), ρεω(x′)=ρω(x′/ε), where ρω is the realization of a random function ρ(ω) , and Aε(x′,ξ)=a(Tx′ /εω,ξ) , the map a(ω,ξ) being measurable in ω and satisfying degenerated structure conditions with weight ρ in ξ. As usual in dimension reduction problems, we focus on the rescaled equations and we prove that under the condition h(ε)/ε→0 , the sequence of solutions of them converges to a limit u0, where u0 is the solution of an (N−1) -dimensional limit problem with homogenized and auxiliary equations.

MSC classification

Secondary: 35B27: Homogenization; equations in media with periodic structure 35J60: Nonlinear elliptic equations 35R60: Partial differential equations with randomness, stochastic partial differential equations 60G57: Random measures 60H25: Random operators and equations
Type
Research Article
Information
LMS Journal of Computation and Mathematics , Volume 15 , May 2012 , pp. 1 - 22
Copyright
Copyright © London Mathematical Society 2012

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