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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

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