Let n ≥ 2 n\ge 2 and Ω ⊂ R n \Omega \subset {{\mathbb{R}}}^{n} be a bounded nontangentially accessible 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 antisymmetric part in Ω \Omega . More precisely, for any given p ∈ ( 2 , ∞ ) p\in \left(2,\infty ) , the authors prove that a weak reverse Hölder inequality with exponent p p implies the global W 1 , p {W}^{1,p} estimate and the global weighted W 1 , q {W}^{1,q} estimate, with q ∈ [ 2 , p ] q\in \left[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 {\rm{BMO}} symmetric part and small BMO {\rm{BMO}} antisymmetric part, respectively, on bounded Lipschitz domains, quasi-convex domains, Reifenberg flat domains, C 1 {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.

中文翻译：

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具有 BMO 反对称部分的椭圆算子 Dirichlet 问题的全局梯度估计

让 n ≥ 2 n\ge 2 和 Ω ⊂ R n \Omega \subset {{\mathbb{R}}}^{n} 是一个有界的非切向可访问域。在本文中，作者研究了具有椭圆对称部分和 BMO 反对称部分的二阶椭圆散度方程的 Dirichlet 边值问题的（加权）全局梯度估计。 Ω \欧米茄 . 更准确地说，对于任何给定的 p ∈ ( 2 , ∞ ) p\in \left(2,\infty ) ，作者证明了具有指数的弱反向 Hölder 不等式 p p 意味着全球 W 1 , p {W}^{1,p} 估计和全局加权 W 1 , q {W}^{1,q} 估计，与 q ∈ [ 2 , p ] q\in \left[2,p] 和一些 Muckenhoupt 权重，用于解决 Dirichlet 边值问题。作为应用，作者建立了一些全局梯度估计，用于求解具有小散度形式的二阶椭圆方程的狄利克雷边值问题。 蒙特利尔银行 {\rm{BMO}} 对称部分和小 蒙特利尔银行 {\rm{BMO}} 反对称部分，分别在有界 Lipschitz 域、准凸域、Reifenberg 平面域上， C 1 {C}^{1} 加权勒贝格空间中的域或（半）凸域。此外，作为进一步的应用，作者分别在（加权）Lorentz 空间、（Lorentz-）Morrey 空间、（Musielak-）Orlicz 空间和可变 Lebesgue 空间中获得了全局梯度估计。即使在 Lebesgue 空间中的全局梯度估计上，本文中获得的结果通过削弱对系数矩阵的假设来改进已知结果。