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

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. 1. Introduction In recent years, there has been an increasing interest in the study of the functionals with variable exponents or nonstandard -growth and the corresponding Sobolev spaces, see for instance [1–8] and the references therein. In particular, the conditions under which functions are dense in have been found. Also, Meyers estimates, which are used in the homogenization process, have been obtained in [6]. 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. Attention is focussed on the homogenization and minimization problems for variational functionals in the framework of Sobolev spaces with nonstandard growth. The material is essentially a review with some new results. -convergence and minimization problems for functionals with periodic and locally periodic rapidly oscillating Lagrangians of -growth with a constant are well studied now, see for instance [9, 10] and the bibliography therein. The works [11–15] (see also [16]) focus on the variational functionals with nonstandard growth conditions. In particular, the homogenization and -convergence problems for Lagrangians with variable rapidly oscillating exponents were considered in [13, 14]. It was shown that the energy minimums and the homogenized Lagrangians in the spaces might depend on the value of (the so-called Lavrentiev phenomenon). For example, such a behavior can be observed for the Lagrangian with a periodic “chess board” exponent and a small parameter . Another