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Existence of a sequence satisfying Cioranescu-Murat conditions in homogenization of Dirichlet problems in perforated domains. (English) Zbl 0870.35013

The paper is concerned with the study of the homogenization problem \[ -\Delta_p u_n=f\quad \text{ in } {\mathcal D}'(\Omega_n),\;u_n\in W_0^{1,p}(\Omega_n), \] where \(\Omega_n\) denotes a sequence of open sets contained in a fixed bounded open set \(\Omega\subset \mathbb{R}^n\), \(p\) is a given number with \(1<p<+\infty\), \(f\) is an element of \(W^{-1,p'}(\Omega)\), and \(\Delta_p\) is the \(p\)-Laplacian operator defined by \(-\Delta_p u=-\text{div }|\nabla u|^{p-2}\nabla u . \) The solutions \(u_n\) are bounded in \(W_0^{1,p}(\Omega)\) and so, there exists a subsequence which converges weakly to a function \(u\) in \(W_0^{1,p}(\Omega)\). The homogenization problem is to find the equation satisfied by the function \(u\) and also to give an approximate representation of the gradient of \(u_n\) in the strong topology of \(L^p(\Omega)\) using the function \(u\) and some explicit auxiliary functions. Originally such a problem is stated and solved by Cioranescu and Murat for classical solutions. Later these results have been extended by Labani and Picard for \(p\)-Laplacians. These results are stated imposing several hypotheses about the sequences \(\Omega_n\). The present paper shows that these conditions can be reduced to the following one: There exists a sequence \(z_n\in W^{1,p}(\Omega)\) which is zero in \(\Omega\setminus\Omega_n\) and which converges weakly to \(1\) in \(W^{1,p}(\Omega)\).
Reviewer: I.Ginchev (Varna)

MSC:

35B27 Homogenization in context of PDEs; PDEs in media with periodic structure
35J65 Nonlinear boundary value problems for linear elliptic equations
35B40 Asymptotic behavior of solutions to PDEs

Keywords:

\(p\)-Laplacian