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Estimations fines des coefficients homogénéisés. (Fine estimations of homogenized coefficients). (French) Zbl 0586.35004
Ennio de Giorgi Colloq., H. Poincaré Inst., Paris 1983, Res. Notes Math. 125, 168-187 (1985).
[For the entire collection see Zbl 0563.00011.]
Second order partial differential elliptic equations, with the matrix A($$\epsilon)$$ of coefficients depending on a parameter $$\epsilon$$, are considered. In the framework of homogenization theory, the conditions are studied in order to obtain the convergence of the matrix A($$\epsilon)$$ as $$\epsilon$$ goes to zero. Using the compensated compactness, the author proves that the set of matrices A($$\epsilon)$$ is compact with respect to the homogenization convergence. Moreover sufficient conditions on the limit matrix A(0) are discussed. Two proofs are furnished by the construction of a sequence of matrix A($$\epsilon)$$ which converges to A(0), in the homogenization sense. One of these proofs uses the explicit calculus of ellipsoidal coordinates.
Reviewer: M.Codegone

##### MSC:
 35B40 Asymptotic behavior of solutions to PDEs 35B45 A priori estimates in context of PDEs 35F15 Boundary value problems for linear first-order PDEs 35J15 Second-order elliptic equations 35B65 Smoothness and regularity of solutions to PDEs