Homogenization of degenerate elliptic equations. (Homogénéisation de problèmes elliptiques dégénérés.) (French. Abridged English version) Zbl 0920.35024

Bellieud, Michel; Bouchitté, Guy

C. R. Acad. Sci., Paris, Sér. I, Math. 327, No. 8, 787-792 (1998).

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