Quantitative Estimates on Periodic Homogenization of Nonlinear Elliptic Operators

Abstract

In this paper, we are interested in the periodic homogenization of quasilinear elliptic equations. We obtain error estimates O(ε 1/2 ) for a C 1,1 domain, and O(ε σ ) for a Lipschitz domain, in which σ ∈ (0, 1/2) is close to zero. Based upon the convergence rates, an interior Lipschitz estimate, as well as a boundary Hölder estimate can be developed at large scales without any smoothness assumption, and these will implies reverse Hölder estimates established for a C 1 domain. By a real method developed by Z.Shen [26], we consequently derive a global W 1,p estimate for 2 ≤ p < ∞. This work may be regarded as an extension of [5,24] to a nonlinear operator, and our results may be extended to the related Neumann boundary problems without any real difficulty.

Quantitative estimates for homogenization of nonlinear elliptic operators in perforated domains

“…A fortunate remedy seems to be a careful extension argument, which just relies on the character of the boundary ∂ω. In general, it is proved to be the fundamental idea for homogenization on perforated domains, such as the extension theorem developed by E. Acervi, V. Piat, G. [39] workable to the present model. In this connection, we would like to address some specific difficulties encountered in the proof of Theorem 1.1, as well as, the related ideas and comments.…”

“…(i). The fact that Y ∩ω ∇N(•, ξ)dy = 0 prevents us from simply repeating the proof used in [39,Lemma 2.3] or [30,Lemma 1] to prove the coercive property of A (see Lemma 2.3), while this property plays a crucial role in the quantitative homogenization theory as we have explained in [38,39] with details. Given its importance, we employ the extension theorem developed in [1, Theorem 2.1] to show a clear proof for this property, inspired by a similar result stated in [32,46].…”

“…In this paragraph, we outline the main idea on error estimates concerning the perforated domains. As developed in the previous framework [38,39] for unperforated domains, consider the two-scale expansion w ε = u ε − v with v = u 0 + εN(y, ϕ), and…”

“…where wε is the extension of w ε given by Lemma 2.12. Compared to the previous work in [38,39], some new tricks have been developed for the term T 1 in the present paper, which do not require any higher regularity assumption on F . In fact, the integrand function in T 1 may be divided into two parts:…”

“…Regarding the remainder terms T 2 , T 3 and T 4 , roughly speaking, one may follow from a similar philosophy summarized in [38,39], and some related challenges have been stated previously. Here the ultimate aim of the computations is reducing the right-hand side of (15) to the so-called "layer" and "co-layer" type estimates…”