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Homogenization of variational functionals with nonstandard growth in perforated domains

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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(|$\nabla$ 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.

    Mathematics Subject Classification: Primary: 35B40; 35J60; 46E35; Secondary: 74Q05; 76M50.


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