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Stochastic-Periodic Homogenization of a Class of Minimization Problems in Orlicz-Sobolev Spaces

Dongho Joseph, Fotso Tachago Joel, Tchinda Takougoum Franck

Abstract

We develop the stochastic two-scale convergence method in the framework of Orlicz-Sobolev spaces, in order to deal with the homogenization of coupled stochastic-periodic problems in such spaces. One fundamental in this topic is the extension of compactness results for this method to the Orlicz setting. For the application, we show that the sequence of minimizers of a class of highly oscillatory minimizations problems involving integral functionals with convex and nonstandard growth integrands, converges to the minimizer of a homogenized problem.

Stochastic-Periodic Homogenization of a Class of Minimization Problems in Orlicz-Sobolev Spaces

Abstract

We develop the stochastic two-scale convergence method in the framework of Orlicz-Sobolev spaces, in order to deal with the homogenization of coupled stochastic-periodic problems in such spaces. One fundamental in this topic is the extension of compactness results for this method to the Orlicz setting. For the application, we show that the sequence of minimizers of a class of highly oscillatory minimizations problems involving integral functionals with convex and nonstandard growth integrands, converges to the minimizer of a homogenized problem.
Paper Structure (16 sections, 16 theorems, 86 equations)

This paper contains 16 sections, 16 theorems, 86 equations.

Table of Contents

  1. Introduction
  2. Our hypotheses
  3. Literature review
  4. Motivation and objectives
  5. Main results
  6. Organization of paper
  7. General notations
  8. Stochastic two-scale convergence in Orlicz-Sobolev spaces
  9. Fundamentals of stochastic-periodic homogenization
  10. Compactness theorem 1: case of Orlicz's spaces
  11. Compactness theorem 2: case of Orlicz-Sobolev spaces
  12. Application to the homogenization of a class of minimization problems
  13. Existence and uniqueness of minimizers for (\ref{['ras2']})
  14. Preliminary convergence results
  15. Regularization
  16. ...and 1 more sections

Key Result

Theorem 1.1

(compactness 1) Let $\Phi \in \Delta_{2}$ be a Young function and let $E$ be a fundamental sequence (see, Section sect3). Then, any bounded sequence $(u_{\epsilon})_{\epsilon\in E}$ in $L^{\Phi}(Q\times\Omega)$ admits a subsequence which is weakly stochastically 2-scale convergent in $L^{\Phi}(Q\tim

Theorems & Definitions (30)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Definition 3.1
  • Remark 3.1
  • Lemma 3.1
  • proof
  • Lemma 3.2
  • proof
  • proof
  • ...and 20 more