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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(|∇ uε | pε (x)-2 ∇ 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 p0(x) and that p0 satisfy certain logarithmic uniform continuity condition, it is shown that uε converges, as ε→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[J]. Networks and Heterogeneous Media, 2010, 5(2): 189-215. doi: 10.3934/nhm.2010.5.189

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  • 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(|∇ uε | pε (x)-2 ∇ 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 p0(x) and that p0 satisfy certain logarithmic uniform continuity condition, it is shown that uε converges, as ε→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.



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    2. Brahim Amaziane, Leonid Pankratov, Homogenization in Sobolev Spaces with Nonstandard Growth: Brief Review of Methods and Applications, 2013, 2013, 1687-9643, 1, 10.1155/2013/693529
    3. Jean Carlos Nakasato, Igor Pažanin, Asymptotic modeling of the current flow system described by the p(x)p(x)‐Laplacian, 2024, 0044-2267, 10.1002/zamm.202400253
    4. Alexander A. Kovalevsky, Approximation in
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