Regularity of the correctors and local gradient estimate of the homogenization for the elliptic equation: linear periodic case

Abstract

$C^{\alpha }$ and $W^{1,\infty }$ estimates for the first-order and second-order correctors in the homogenization are presented based on the translation invariant and Li-Vogelius's gradient estimate for the second order linear elliptic equation with piecewise smooth coefficients. If the data are smooth enough, the error of the first-order expansion for piecewise smooth coefficients is locally $O(\epsilon )$ in the Hölder norm; it is locally $O(\epsilon )$ in $W^{1,\infty }$ when coefficients are Lipschitz continuous. It can be partly extended to the nonlinear parabolic equation.

arXiv comment4 pages; a part of the first author's doctoral dissertation

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