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

Abstract

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 W m−1,p0 with p 0 = 2d d−1 in a bounded Lipschitz domain in R d as well as the uniform large-scale interior C m−1,1 estimate. With additional smoothness assumptions, the uniform interior C m−1,1 , W m,p and C m−1,α estimates are also obtained. As applications of the regularity estimates, we establish asymptotic expansions for fundamental solutions.

Periodic homogenization of elliptic systems with stratified structure

“…where v k,0 is the unique solution of L A k 0 (v k,0 ) = f in Ω and v k,0 = u 0 on ∂Ω. Moreover, by using (26), one can verify that for any x,…”

Periodic homogenization of Green’s functions for Stokes systems

“…In [26] and [30], the method of Bloch waves was used to study the asymptotic expansions of the fundamental solutions and heat kernels, respectively. In [6], the asymptotic expansions of the fundamental solutions for elliptic operator L ε were obtained via the uniform regularity theory established in [3,4]; and most recently, the results were extended to the higher order elliptic systems in [25] and to parabolic equations in [13]. For elliptic systems in a bounded domain, the asymptotic expansion for the Poisson kernel was obtained in [5], using the Dirichlet correctors.…”

Uniform boundary estimates in homogenization of higher-order elliptic systems

“…In [39,40], some interesting two-parameter resolvent estimates were established in homogenization of general higherorder elliptic systems with periodic coefficients in bounded C 2m domains. Meanwhile, in [28,45] we investigated the sharp O(ε) convergence rate in periodic and almost periodic homogenization of higher-order elliptic systems in Lipschitz domains. Particularly, under the assumptions that A is symmetric and u 0 ∈ H m+1 ( ), the optimal O(ε) convergence rate was obtained in W m−1,q 0 ( ), q 0 = 2d/(d − 1) in [28].…”

“…Meanwhile, in [28,45] we investigated the sharp O(ε) convergence rate in periodic and almost periodic homogenization of higher-order elliptic systems in Lipschitz domains. Particularly, under the assumptions that A is symmetric and u 0 ∈ H m+1 ( ), the optimal O(ε) convergence rate was obtained in W m−1,q 0 ( ), q 0 = 2d/(d − 1) in [28]. Moreover, the uniform interior W m, p and C m−1,1 estimates were also established therein.…”