International Journal of Differential Equations
Volume 2013 (2013), Article ID 693529, 16 pages
Research Article

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

1Laboratoire de Mathėmatiques et de leurs Applications, CNRS-UMR 5142, UNIV Pau & Pays Adour, Avenue de l’Universitė, 64000 Pau, France
2Department of Mathematics, B. Verkin Institute for Low Temperature Physics and Engineering, 47, Avenue Lenin, Kharkov 61103, Ukraine

Received 12 July 2012; Accepted 22 January 2013

Academic Editor: Julio Rossi

Copyright © 2013 Brahim Amaziane and Leonid Pankratov. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


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.