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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.

##### MSC:
 35B37 PDE in connection with control problems (MSC2000) 35J70 Degenerate elliptic equations
##### Keywords:
non-local effective law; linear elasticity
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