# American Institute of Mathematical Sciences

June  2010, 5(2): 189-215. doi: 10.3934/nhm.2010.5.189

## Homogenization of variational functionals with nonstandard growth in perforated domains

 1 Laboratoire de Mathématiques et de leurs Applications, CNRS-UMR 5142, Université de Pau, Av. de l’Université, 64000 Pau, France 2 Department of Mathematics, B.Verkin Institute for Low Temperature Physics and Engineering, 47, av. Lenin, 61103, Kharkov, Ukraine 3 Narvik University College, Postbox 385, Narvik, 8505, Norway

Received  November 2009 Revised  February 2010 Published  May 2010

The aim of the paper is to study the asymptotic behavior of solutions to a Neumann boundary value problem for a nonlinear elliptic equation with nonstandard growth condition of the form

-div(|$\nabla$ uε | pε (x)-2 $\nabla$ uε )+ (| uε | pε (x)-2 uε = f(x)

in a perforated domain Ωε , ε being a small parameter that characterizes the microscopic length scale of the microstructure. Under the assumption that the functions pε(x) converge uniformly to a limit function $p_0(x)$ and that $p_0$ satisfy certain logarithmic uniform continuity condition, it is shown that uε converges, as ε$\to 0$, to a solution of homogenized equation whose coefficients are calculated in terms of local energy characteristics of the domain Ωε . This result is then illustrated with periodic and locally periodic examples.

Citation: Brahim Amaziane, Leonid Pankratov, Andrey Piatnitski. Homogenization of variational functionals with nonstandard growth in perforated domains. Networks & Heterogeneous Media, 2010, 5 (2) : 189-215. doi: 10.3934/nhm.2010.5.189
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