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Homogenization of degenerate elliptic equations. (Homogénéisation de problèmes elliptiques dégénérés.) (French. Abridged English version) Zbl 0920.35024
Summary: We study the homogenization of degenerate elliptic equations like: \[ -\text{div}(a_\varepsilon\nabla u)= f\quad\text{on }\Omega, \] \[ u= u_0\quad\text{on }\Gamma_0\subset\partial \Omega,\quad {\partial u\over\partial n}= 0\quad\text{on }\partial\Omega\setminus \Gamma_0, \] where the coefficient \(a_\varepsilon\) is \(\varepsilon\)-periodic, takes values of order 1 on a subset \(T_\varepsilon\subset \Omega\) (fiber structure) and of order \(\varepsilon^2\) on \(\Omega\setminus \Gamma_\varepsilon\). We obtain a non-local effective law which cannot be derived from the theory of bounds by assuming that \(a_\varepsilon\) is lower bounded by a fixed parameter which tends to \(0\) afterwards. Some variants of our results are given in the case of linear elasticity.

35B37 PDE in connection with control problems (MSC2000)
35J70 Degenerate elliptic equations
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