Let $n\ge 2$ and Ω be a bounded Lipschitz domain of ${\mathbb{R}}^n$. In this article, the authors investigate global (weighted) estimates for the gradient of solutions to Robin boundary value problems of second-order elliptic equations of divergence form with real-valued, bounded, measurable coefficients on Ω. More precisely, let $p\in (n/(n-1),\infty )$. Using a real-variable argument, the authors obtain two necessary and sufficient conditions for $W^{1,p}$ estimates of solutions to Robin boundary value problems, respectively, in terms of a weak reverse Hölder inequality with exponent p or weighted $W^{1,q}$ estimates of solutions with $q\in (n/(n-1),p]$ and some Muckenhoupt weights. As applications, the authors establish some global regularity estimates for solutions to Robin boundary value problems of second-order elliptic equations of divergence form with small BMO coefficients, respectively, on bounded Lipschitz domains, $C^1$ domains or (semi-)convex domains, in the scale of weighted Lebesgue spaces, via some quite subtle approach which is different from the existing ones and, even when $n=3$ in case of bounded $C^1$ domains, also gives an alternative correct proof of some known result under an additional assumption. By this and some technique from harmonic analysis, the authors further obtain the global regularity estimates, respectively, in Morrey spaces, (Musielak–)Orlicz spaces, and variable Lebesgue spaces.

让 $n\ge 2$ 和 Ω 是一个有界的 Lipschitz 域 ${\mathbb{电阻}}^n$. 在本文中，作者研究了在 Ω 上具有实值、有界、可测量系数的散度形式的二阶椭圆方程的 Robin 边值问题解的梯度的全局（加权）估计。更准确地说，让$磷\in (n/(n-1),\infty )$. 使用实变量论证，作者获得了两个充分必要条件${宽}^{1,磷}$分别根据指数为p或加权的弱反向 Hölder 不等式对 Robin 边值问题的解的估计${宽}^{1,q}$ 解决方案的估计 $q\in (n/(n-1),磷]$和一些 Muckenhoupt 重量。作为应用，作者分别为有界 Lipschitz 域上具有小 BMO 系数的散度形式的二阶椭圆方程的 Robin 边值问题的解建立了一些全局正则性估计，$C^1$ 域或（半）凸域，在加权 Lebesgue 空间的尺度上，通过一些与现有方法不同的非常微妙的方法，即使当 $n=3$ 在有界的情况下 $C^1$域，还提供了在附加假设下某些已知结果的替代正确证明。通过这个和调和分析的一些技术，作者进一步获得了分别在 Morrey 空间、(Musielak-)Orlicz 空间和变量 Lebesgue 空间中的全局正则性估计。