Homogenization in Sobolev Spaces with Nonstandard Growth: Brief Review of Methods and Applications

Brahim Amaziane,Leonid Pankratov

doi:10.1155/2013/693529

Abstract

We review recent results on the homogenization in Sobolev spaces with variable exponents. In particular, we are dealing with the Γ-convergence of variational functionals with rapidly oscillating coefficients, the homogenization of the Dirichlet and Neumann variational problems in strongly perforated domains, as well as double porosity type problems. The growth functions also depend on the small parameter characterizing the scale of the microstructure. The homogenization results are obtained by the method of local energy characteristics. We also consider a parabolic double porosity type problem, which is studied by combining the variational homogenization approach and the two-scale convergence method. Results are illustrated with periodic examples, and the problem of stability in homogenization is discussed.

Highlights

In recent years, there has been an increasing interest in the study of the functionals with variable exponents or nonstandard p(x)-growth and the corresponding Sobolev spaces, see for instance [1,2,3,4,5,6,7,8] and the references therein
Let us mention that such partial differential equations arise in many engineering disciplines, such as electrorheological fluids, non-Newtonian fluids with thermoconvective effects, and nonlinear Darcy flow of compressible fluids in heterogeneous porous media, see for instance [1]
This paper discusses problems of homogenization in Sobolev spaces with variable exponents

Summary

Introduction

There has been an increasing interest in the study of the functionals with variable exponents or nonstandard p(x)-growth and the corresponding Sobolev spaces, see for instance [1,2,3,4,5,6,7,8] and the references therein. Γ-convergence and minimization problems for functionals with periodic and locally periodic rapidly oscillating Lagrangians of p-growth with a constant p are well studied see for instance [9, 10] and the bibliography therein. It was shown that the energy minimums and the homogenized Lagrangians in the spaces W1,r might depend on the value of r (the so-called Lavrentiev phenomenon) Such a behavior can be observed for the Lagrangian |∇u|p(x/ε) with a periodic “chess board” exponent p(y) and a small parameter ε > 0. It is based on the derivation of the lim-inf and lim-sup estimates for the variational functional under consideration along with the assumptions on the behavior of the local energy characteristics. The main result of the section (Theorem 23) is obtained by the method of local energy characteristics. The main results of the section are obtained by combining the two-scale convergence method and the variational homogenization approach

Sobolev Spaces with Variable Exponents

Definitions

Homogenization of the Dirichlet Problem and Related Questions

Nonlocal in Perforated

Homogenization of the Neumann Problem