Homogenization of elliptic systems with periodic coefficients: Weighted $L^p$ and $L^{\mathrm{\infty }}$ estimates for asymptotic remainders

S. A. Nazarov

Algebra i analiz, Tome 18 (2006) no. 2, pp. 117-166.

Résumé

The difference between the fundamental matrix for a second order selfadjoint elliptic system with sufficiently smooth periodic coefficients and the fundamental matrix for the corresponding homogenized system in ${\mathbb{R}}^n$ is shown to decay as $O(1+|x{|}^{1-n}$) at infinity, $n\ge 2$. As a consequence, weighted $L^p$ and $L^{\mathrm{\infty }}$ estimates are obtained for the difference $u^{\epsilon }-u^0$ of the solutions of a system with rapidly oscillating periodic coefficients and the homogenized system in ${\mathbb{R}}^n$ with right-hand side belonging to an appropriate weighted $L^p$-class in ${\mathbb{R}}^n$.

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