Uniform regularity estimates in homogenization theory of elliptic systems with lower order terms on the Neumann boundary problem

Abstract

In this paper, we mainly employed the idea of the previous paper [36] to study the sharp uniform W 1,p estimates with 1 < p ≤ ∞ for more general elliptic systems with the Neumann boundary condition on a bounded C 1,η domain, arising in homogenization theory. Based on the skills developed by Z. Shen in [29] and by T. Suslina in [33,34], we also established the L 2 convergence rates on a bounded C 1,1 domain and a Lipschitz domain, respectively. Here we found a "rough" version of the first order correctors (see (1.12)), which can unify the proof in [29] and [34]. It allows us to skip the corresponding convergence results on R d that are the preconditions in [33,34]. Our results can be regarded as an extension of [24] developed by C. Kenig, F. Lin, Z. Shen, as well as of [34] investigated by respectively. Then the result of [24, Theorem 1.2] can be directly applied to (D 2 ). Due to the same reason as explained in [36], we need to derivewhich follows from the decay estimates of Neumann matrixes defined in [24, pp.916] (see Theorem 4.2), as well as ∇Ψ ε,0 C 0,σ 1 (Ω) = O(ε −σ 1 ) and ∇u ε C 0,σ 1 (Ω) = O(ε −σ 2 ) as ε → 0, which are the main conclusions of Corollary 4.6 and Lemma 4.8, where 0 < σ 1 < σ 2 < 1 are independent of ε. The above two estimates together with (1.6) guarantee that the right hand side of (D 2 ) can be uniformly bounded by the given data in Theorem 1.2. Note that the right hand side of (D 2 ) which involves div(f ) is, as a matter of fact, more general than that in [24, Theorem 1.2]. We find a simple way inspired by [31] to derive the uniform Lipschitz estimate for the weak solution u ε to L ε (u ε ) = div(f ) in Ω and ∂u ε /∂ν ε = −n · f on ∂Ω. The key ingredient is to construct the auxiliary functions {v ε,k } d k=1 , which satisfy L ε (v ε,k ) = 0 in Ω, ∂v ε,k /∂ν ε = −n k I on ∂Ω

Convergence Rates for General Elliptic Homogenization Problems in Lipschitz Domains

“…Hence, the task is reduced to track the constant, and we repeat using Caccioppoli's inequality to send the factor "λ − 1 2 " to the coefficients V, B and c. The details may be found in the proof Theorem 1.1. Generally speaking, the main ideas in Theorem 1.1 is using Caccioppoli's inequality stated above (or see Lemma 2.7) to improve the related estimates in [37] such that C(µ, κ, λ, m, d, p A VMO ) (1.6), C(µ, κ, λ, m, d, τ ) (1.9).…”

Optimal Boundary Estimates for Stokes Systems in Homogenization Theory

“…If the reader interests in the boundary estimates, we highly recommend Z. Shen's elegant work [17]. The quantitative homogenization has been extensively studied, we refer the reader to [1,2,3,4,7,8,9,10,13,18,19,24,25] and their references therein.…”

Uniform boundary estimates in homogenization of higher-order elliptic systems

“…Note that although the result presented in Theorem 1.2 focuses on the case p ≥ 2, by a standard duality argument, it still holds for 1 < p < 2. We also mention that the uniform W 1, p estimates in the homogenization of second-order elliptic systems have been studied largely, see e.g., [14,15,33,43,44]. Theorem 1.2 generalizes the uniform W 1, p estimates for second-order elliptic systems to higher-order elliptic systems.…”