Abstract Let n ≥ 2 and Ω be a bounded Lipschitz domain in R n . In this article, the authors investigate global (weighted) norm estimates for the gradient of solutions to Neumann boundary value problems of second order elliptic equations of divergence form with real-valued, bounded, measurable coefficients in Ω. More precisely, for any given p ∈ ( 2 , ∞ ) , two necessary and sufficient conditions for W 1 , p estimates of solutions to Neumann boundary value problems, respectively, in terms of a weak reverse Holder inequality with exponent p or weighted W 1 , q estimates of solutions with q ∈ [ 2 , p ] and some Muckenhoupt weights, are obtained. As applications, for any given p ∈ ( 1 , ∞ ) and ω ∈ A p ( R n ) (the class of Muckenhoupt weights), the authors establish weighted W ω 1 , p estimates for solutions to Neumann boundary value problems of second order elliptic equations of divergence form with small BMO coefficients on bounded (semi-)convex domains. As further applications, the global gradient estimates are obtained, respectively, in (weighted) Lorentz spaces, (Lorentz–)Morrey spaces, (weighted) Orlicz spaces, and variable Lebesgue spaces.

中文翻译：

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Lipschitz 和（半）凸域中具有 Neumann 边界条件的椭圆问题的加权梯度估计

摘要 令n ≥ 2 且Ω 为R n 中的有界Lipschitz 域。在本文中，作者研究了具有实值、有界、可测系数的二阶椭圆方程的 Neumann 边值问题的解梯度的全局（加权）范数估计，其单位为 Ω。更准确地说，对于任何给定的 p ∈ ( 2 , ∞ ) ，W 1 的两个必要和充分条件，分别是 Neumann 边界值问题的解的 p 估计值，根据弱反向 Holder 不等式，指数为 p 或加权 W 1 , q 对 q ∈ [ 2 , p ] 和一些 Muckenhoupt 权重的解的估计得到了。作为应用，对于任何给定的 p ∈ ( 1 , ∞ ) 和 ω ∈ A p ( R n )（Muckenhoupt 权重的类），作者建立加权 W ω 1 ，p 对有界（半）凸域上具有小 BMO 系数的发散形式的二阶椭圆方程的 Neumann 边值问题的解的估计。作为进一步的应用，全局梯度估计分别在（加权）洛伦兹空间、（洛伦兹-）莫雷空间、（加权）Orlicz 空间和可变 Lebesgue 空间中获得。