International Journal of Differential Equations

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

Brahim Amaziane and Leonid Pankratov

Full-text: Open access

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.

Article information

Source
Int. J. Differ. Equ., Volume 2013 (2013), Article ID 693529, 16 pages.

Dates
Received: 12 July 2012
Accepted: 22 January 2013
First available in Project Euclid: 20 January 2017

Permanent link to this document
https://projecteuclid.org/euclid.ijde/1484881338

Digital Object Identifier
doi:10.1155/2013/693529

Mathematical Reviews number (MathSciNet)
MR3046764

Zentralblatt MATH identifier
1270.35054

Citation

Amaziane, Brahim; Pankratov, Leonid. Homogenization in Sobolev Spaces with Nonstandard Growth: Brief Review of Methods and Applications. Int. J. Differ. Equ. 2013 (2013), Article ID 693529, 16 pages. doi:10.1155/2013/693529. https://projecteuclid.org/euclid.ijde/1484881338


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