Homogenization of variational functionals with nonstandard growth in perforated domains

Brahim Amaziane (1) , Leonid Pankratov (1) , Andrey Piatnitski (2)

1 LMAP - Laboratoire de Mathématiques et de leurs Applications [Pau]
2 Narvik University College

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 (vertical bar del u(epsilon)vertical bar(p epsilon(x)-2) del u(epsilon)) + vertical bar del u(epsilon)vertical bar(p epsilon(x)-2) u(epsilon) = f (x) in a perforated domain Omega(epsilon), epsilon being a small parameter that characterizes the microscopic length scale of the microstructure. Under the assumption that the functions p(epsilon)(x) converge uniformly to a limit function p(0)(x) and that p0 satisfy certain logarithmic uniform continuity condition, it is shown that u(epsilon) converges, as epsilon -> 0, to a solution of homogenized equation whose coefficients are calculated in terms of local energy characteristics of the domain Omega(epsilon). This result is then illustrated with periodic and locally periodic examples.

Domaines: Analyse numérique [math.NA]

Brahim Amaziane, Leonid Pankratov, Andrey Piatnitski. Homogenization of variational functionals with nonstandard growth in perforated domains. Networks and Heterogeneous Media, 2010, 5 (2), pp.189-215. ⟨10.3934/nhm.2010.5.189⟩. ⟨hal-00867192⟩