Global Gradient Estimates for Dirichlet Problems of Elliptic Operators with a BMO Anti-Symmetric Part

Abstract

Let n ≥ 2 and Ω ⊂ R n be a bounded NTA domain. In this article, the authors investigate (weighted) global gradient estimates for Dirichlet boundary value problems of second order elliptic equations of divergence form with an elliptic symmetric part and a BMO anti-symmetric part in Ω. More precisely, for any given p ∈ (2, ∞), the authors prove that a weak reverse Hölder inequality with exponent p implies the global W 1,p estimate and the global weighted W 1,q estimate, with q ∈ [2, p] and some Muckenhoupt weights, of solutions to Dirichlet boundary value problems. As applications, the authors establish some global gradient estimates for solutions to Dirichlet boundary value problems of second order elliptic equations of divergence form with small BMO symmetric part and small BMO anti-symmetric part, respectively, on bounded Lipschitz domains, quasi-convex domains, Reifenberg flat domains, C 1 domains, or (semi-)convex domains, in weighted Lebesgue spaces. Furthermore, as further applications, the authors obtain the global gradient estimate, respectively, in (weighted) Lorentz spaces, (Lorentz-)Morrey spaces, (Musielak-)Orlicz spaces, and variable Lebesgue spaces. Even on global gradient estimates in Lebesgue spaces, the results obtained in this article improve the known results via weakening the assumption on the coefficient matrix.

Heat Kernels and Hardy Spaces on Non-Tangentially Accessible Domains with Applications to Global Regularity of Inhomogeneous Dirichlet Problems

“…Furthermore, a (semi-)convex domain is also a (γ, σ, R) quasi-convex domain for any γ ∈ (0, 1), some σ ∈ (0, 1), and some R ∈ (0, ∞) (see, for instance, [69]). (iii) On NTA domains, Lipschitz domains, quasi-convex domains, Reifenberg flat domains, C 1 domains, and (semi-)convex domains, we have the following relations (see, for instance, [38,40,44,64,67,69]). Next, we recall the definition of the (γ, R)-BMO condition (see, for instance, [11]).…”

“…It is worth pointing out that convex domains of R n are semi-convex domains and (semi-)convex domains are special cases of Lipschitz domains (see, for instance, [48,49,67]). Furthermore, a (semi-)convex domain is also a (γ, σ, R) quasi-convex domain for any γ ∈ (0, 1), some σ ∈ (0, 1), and some R ∈ (0, ∞) (see, for instance, [69]). (iii) On NTA domains, Lipschitz domains, quasi-convex domains, Reifenberg flat domains, C 1 domains, and (semi-)convex domains, we have the following relations (see, for instance, [38,40,44,64,67,69]).…”

“…Via replacing [69,Lemma 4.3] by the following Lemma 3.8, using a real-variable argument for L p (Ω) estimates, which was essentially established by Shen [58,Theorem 3.4] (see also [57,Theorem 4.2.6] and [59, Theorem 3.3]), and then repeating the proof of [69, Theorem 1.10], we can prove Lemma 3.7; we omit the details here. Lemma 3.8.…”

“…Moreover, when Ω is a bounded semi-convex domain of R n (see, for instance, [48,49] or Remark 2.5(ii) below for the details), and L D := −∆ D with ∆ D being the Laplace operator with the Dirichlet boundary condition on Ω, Theorem 1.1(ii) was obtained in [26,Lemma 2.7]. Recall that the bounded semi-convex domain Ω is a (γ, σ, R) quasi-convex domain for any γ ∈ (0, 1), some σ ∈ (0, 1), and some R ∈ (0, ∞) (see, for instance, [69]). Thus, Theorem 1.1(ii) essentially improves [26,Lemma 2.7] by weakening the assumptions on both the matrix A and the domain Ω.…”