Convergence rates and interior estimates in homogenization of higher order elliptic systems

This paper is concerned with the quantitative homogenization of 2m-order elliptic systems with bounded measurable, rapidly oscillating periodic coefficients. We establish the sharp O(ε) convergence rate in Wm−1,p0 with p0=2dd−1 in a bounded Lipschitz domain in Rd as well as the uniform large-scale interior Cm−1,1 estimate. With additional smoothness assumptions, the uniform interior Cm−1,1, Wm,p and Cm−1,α estimates are also obtained. As applications of the regularity estimates, we establish asymptotic expansions for fundamental solutions.