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An extension theorem from connected sets, and homogenization in general periodic domains. (English) Zbl 0779.35011

This paper deals with the homogenization problem for an elliptic equation in a periodic domain. It consists of the study of the behaviour of the solutions when the period goes to zero. The authors consider the following family of PDEs: \[ -\Delta u_ \varepsilon+u_ \varepsilon=g \text{ in } \Omega\cap E_ \varepsilon,\qquad \partial u_ \varepsilon/ \partial n_ \varepsilon=0 \text{ in } \partial(\Omega\cap E_ \varepsilon) \] (\(\Omega\subseteq\mathbb{R}^ n\) is a bounded open set with Lipschitz boundary, \(g\in L_ 2(\Omega)\), \(E\) is an arbitrary periodic open subset of \(\mathbb{R}^ n\) with Lipschitz boundary, \(E_ \varepsilon= \varepsilon E\) denotes the \(\varepsilon\)-homothetic set, \(n_ \varepsilon\) is the outward unit normal to \(\partial(\Omega\cap E_ \varepsilon)\)). Main result: Under the only additional assumption that \(E\) is connected, the authors give detailed proofs that there exists an extension \(\widetilde u_ \varepsilon\) of \(u_ \varepsilon\) to the whole of \(\Omega\), such that \(\{\widetilde u_ \varepsilon\}\) converges to the solution \(u\) of the problem: \[ - \sum^ n_{i,j=1} \alpha_{ij} D_ i D_ j u+u=q \text{ in } \Omega, \qquad \partial u/\partial n=0 \text{ in } \partial\Omega \] where \((\alpha_{ij})\) is the positive symmetric matrix defined by \[ \sum^ n_{i,j=1} \alpha_{ij} \xi_ i \xi_ j= \inf\left\{ {1\over {| Q\cap E|}} \int_{Q\cap E} | Du(x)+\xi|^ 2 dx:\;u \text{ is \(Q\)-periodic, } u\in C^ 1(\mathbb{R}^ n) \right\} \] for every \(\xi\in \mathbb{R}^ n\), where \(Q\) is the periodicity cell for \(E\).

MSC:

35B27 Homogenization in context of PDEs; PDEs in media with periodic structure
74E05 Inhomogeneity in solid mechanics
35J60 Nonlinear elliptic equations
Full Text: DOI

References:

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