Abstract
This paper concerns with the quantitative homogenization of second-order elliptic systems with periodic stratified structure in Lipschitz domains. Under the symmetry assumption on coefficient matrix, the sharp \begin{document}$ O(\varepsilon) $\end{document} -convergence rate in \begin{document}$ L^{p_0}(\Omega) $\end{document} with \begin{document}$ p_0 = \frac{2d}{d-1} $\end{document} is obtained based on detailed discussions on stratified functions. Without the symmetry assumption, an \begin{document}$ O(\varepsilon^\sigma) $\end{document} -convergence rate is also derived for some \begin{document}$ \sigma by the Meyers estimate. Based on this convergence rate, we establish the uniform interior Lipschitz estimate. The uniform interior \begin{document}$ W^{1, p} $\end{document} and Holder estimates are also obtained by the real variable method.
Highlights
Let Ω be a bounded Lipschitz domain in Rd, d ≥ 2
Another scheme for large-scale uniform regularity estimates was formulated in [3] and further developed in [2, 29]. It is based on convergence rates and is effective for both Lipschitz and Holder estimates
The matrix A on Ω can be extended onto Rd preserving the Lipschitz property, the boundedness property and the ellipticity condition
Summary
Let Ω be a bounded Lipschitz domain in Rd, d ≥ 2. Homogenization, periodic coefficients, stratified structure, convergence rates, uniform regularity estimates. Another scheme for large-scale uniform regularity estimates was formulated in [3] and further developed in [2, 29] It is based on convergence rates and is effective for both Lipschitz and Holder estimates. We derive the (uniform) local Lipschitz estimates in Theorem 1.2, based on a weaker L2-convergence rate obtained from the Meyers estimate without the symmetry assumption. Note that by the inverse function theorem, diam(U ) depends only on the continuity modulus of ∇ρ and ∇(ρ−1) L∞(Ω). Let Ω be a bounded Lipschitz domain in Rd. Suppose ρ ∈ C1(Ω; Rn) satisfies (5) and h(y) : Rn → R is 1-periodic.
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