Convergence Rates for General Elliptic Homogenization Problems in Lipschitz Domains

Abstract

In terms of layer potential methods, this paper is devoted to study the L 2 boundary value problems for nonhomogeneous elliptic operators with rapidly oscillating coefficients in a periodic setting. Under a low regularity assumption on the coefficients, we establish the solvability for Dirichlet, regular and Neumann problems in a bounded Lipschitz domain, as well as, the uniform nontangential maximal function estimates and square function estimates. The main difficulty is reflected in two aspects: (i) we can not treat the lower order terms as a compact perturbation to the leading term due to the low regularity assumption; (ii) the nonhomogeneous systems do not possess a scaling-invariant property in general. Although this work may be regarded as a follow-up to C. Kenig and Z. Shen's in [21], we make an effort to find a clear way of how to handle the nonhomogeneous operators by using the known results of the homogenous ones. Also, we mention that the periodicity condition plays a key role in the scaling-invariant estimates.

Boundary value problems in Lipschitz domains for equations with lower order coefficients

“…We will now compare differences of Green's functions for the full equation and the fundamental solution when the lower order coefficients vanish. We remark that similar estimates appear in [XZZ18], but the authors compare with the fundamental solution for fixed coefficients (Lemma 4.6). Assume that B = B 10ρ is a ball of radius 10ρ for ρ < 1 16 and let A ∈ M B (λ, α, τ ).…”

“…We remark that a similar estimate to the one we will show can be found in Section 4.1 of [XZZ18]. However, we carry out the proof for the sake of completeness, and in order to show how to extend this estimate in the case of non-symmetric matrices.…”

“…Towards the direction of homogenization of elliptic boundary value problems for equations we refer to [KS11a], and to [KS11b] for elliptic systems (as well as their introductions for more references). A more recent work which allows lower order coefficients is also the paper [XZZ18], in which the authors assume that b, c are Hölder continuous, d is bounded, and the bilinear form for L is coercive. Solvability results for operators L ε in homogenization subsume the analogous results on solvability for L = L 1 , but we believe that the problems we consider in this paper have not been treated before.…”

Convergence rates on periodic homogenization of p-Laplace type equations

“…Comment 4. In both periodic and aperiodic settings, error estimates have always been a hot topic in homogenization theory (see for example [1,2,10,14,16,17,21,24,25,27,28,29,31,32] and their references therein for more details), while there is few contributions in this field concerning p-Laplace type equations, to the authors' best acknowledge. In fact, there seems to be an interesting problem: except of the special cases in Section 5, what kind of conditions can guarantee the homogenized operator L 0 admits the structure (1.7).…”

“…26, Lemma 2.2] and[28, Lemma 3.3], which is based upon the characterization of W 1,p functions, i.e., f(· − εy) − f (·) L p (Σ 2ε ) ≤ ε ∇f L p (Σε) , where |y| ≤ 1.This together with Minkowskis inequality consequently leads to the stated estimate (2.43). So, it is crucial in applications that the constant C is independent of the size of domain.…”