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

Abstract

<p style='text-indent:20px;'>Let <inline-formula><tex-math id="M1">\begin{document}$ n\ge2 $\end{document}</tex-math></inline-formula> and <inline-formula><tex-math id="M2">\begin{document}$ \Omega\subset\mathbb{R}^n $\end{document}</tex-math></inline-formula> be a bounded NTA domain. In this article, the authors study (weighted) global regularity estimates for Neumann boundary value problems of second-order elliptic equations of divergence form with coefficients consisting of both an elliptic symmetric part and a BMO anti-symmetric part in <inline-formula><tex-math id="M3">\begin{document}$ \Omega $\end{document}</tex-math></inline-formula>. Precisely, for any given <inline-formula><tex-math id="M4">\begin{document}$ p\in(2,\infty) $\end{document}</tex-math></inline-formula>, via a weak reverse Hölder inequality with the exponent <inline-formula><tex-math id="M5">\begin{document}$ p $\end{document}</tex-math></inline-formula>, the authors give a sufficient condition for the global <inline-formula><tex-math id="M6">\begin{document}$ W^{1,p} $\end{document}</tex-math></inline-formula> estimate and the global weighted <inline-formula><tex-math id="M7">\begin{document}$ W^{1,q} $\end{document}</tex-math></inline-formula> estimate, with <inline-formula><tex-math id="M8">\begin{document}$ q\in[2,p] $\end{document}</tex-math></inline-formula> and some Muckenhoupt weights, of solutions to Neumann boundary value problems in <inline-formula><tex-math id="M9">\begin{document}$ \Omega $\end{document}</tex-math></inline-formula>. As applications, the authors further obtain global regularity estimates for solutions to Neumann boundary value problems of second-order elliptic equations of divergence form with coefficients consisting of both a small <inline-formula><tex-math id="M10">\begin{document}$ \mathrm{BMO} $\end{document}</tex-math></inline-formula> symmetric part and a small <inline-formula><tex-math id="M11">\begin{document}$ \mathrm{BMO} $\end{document}</tex-math></inline-formula> anti-symmetric part, respectively, in bounded Lipschitz domains, quasi-convex domains, Reifenberg flat domains, <inline-formula><tex-math id="M12">\begin{document}$ C^1 $\end{document}</tex-math></inline-formula> 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.</p>