Homogenization of variational functionals with nonstandard growth in perforated domains

Brahim Amaziane; Leonid Pankratov; Andrey Piatnitski; Brahim Amaziane; Leonid Pankratov; Andrey Piatnitski

doi:10.3934/nhm.2010.5.189

Networks and Heterogeneous Media

2010, Volume 5, Issue 2: 189-215. doi: 10.3934/nhm.2010.5.189

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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
2.
Department of Mathematics, B.Verkin Institute for Low Temperature Physics and Engineering, 47, av. Lenin, 61103, Kharkov
3.
Narvik University College, Postbox 385, Narvik, 8505

Received: 01 November 2009 Revised: 01 February 2010
Primary: 35B40; 35J60; 46E35; Secondary: 74Q05; 76M50.

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(|$\nabla$ u^ε | ^p_ε(x)-2 $\nabla$ u^ε )+ (| u^ε | ^p_ε(x)-2 u^ε= f(x)

in a perforated domain Ω^ε , ε being a small parameter that characterizes the microscopic length scale of the microstructure. Under the assumption that the functions p_ε(x) converge uniformly to a limit function $p_0(x)$ and that $p_0$ satisfy certain logarithmic uniform continuity condition, it is shown that u^ε converges, as ε$ \to 0$, to a solution of homogenized equation whose coefficients are calculated in terms of local energy characteristics of the domain Ω^ε. This result is then illustrated with periodic and locally periodic examples.
- Homogenization,
- perforated domains,
- variational problem,
- nonstandard growth condition
Citation: Brahim Amaziane, Leonid Pankratov, Andrey Piatnitski. Homogenization of variational functionals with nonstandard growth in perforated domains[J]. Networks and Heterogeneous Media, 2010, 5(2): 189-215. doi: 10.3934/nhm.2010.5.189

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Abstract

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(|$\nabla$ u^ε | ^p_ε(x)-2 $\nabla$ u^ε )+ (| u^ε | ^p_ε(x)-2 u^ε= f(x)

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