Global regularity estimates for Neumann problems of elliptic operators with coefficients having a BMO anti-symmetric part in NTA domains

Let \begin{document}$ n\ge2 $\end{document} and \begin{document}$ \Omega\subset\mathbb{R}^n $\end{document} be a bounded NTA domain. In this article, the authors study (weighted) global regularity estimates for Neumann boundary value problems of second-order elliptic equations of divergence form with coefficients consisting of both an elliptic symmetric part and a BMO anti-symmetric part in \begin{document}$ \Omega $\end{document} . Precisely, for any given \begin{document}$ p\in(2,\infty) $\end{document} , via a weak reverse Hölder inequality with the exponent \begin{document}$ p $\end{document} , the authors give a sufficient condition for the global \begin{document}$ W^{1,p} $\end{document} estimate and the global weighted \begin{document}$ W^{1,q} $\end{document} estimate, with \begin{document}$ q\in[2,p] $\end{document} and some Muckenhoupt weights, of solutions to Neumann boundary value problems in \begin{document}$ \Omega $\end{document} . As applications, the authors further obtain global regularity estimates for solutions to Neumann boundary value problems of second-order elliptic equations of divergence form with coefficients consisting of both a small \begin{document}$ \mathrm{BMO} $\end{document} symmetric part and a small \begin{document}$ \mathrm{BMO} $\end{document} anti-symmetric part, respectively, in bounded Lipschitz domains, quasi-convex domains, Reifenberg flat domains, \begin{document}$ C^1 $\end{document} domains, or (semi-)convex domains, in weighted Lebesgue spaces. The results given in this article improve the known results by weakening the assumption on the coefficient matrix.