$ W^{1,p} $ estimates for elliptic systems on composite material with almost partially BMO coefficients

Abstract

<p style='text-indent:20px;'>In this paper, we establish uniform <inline-formula><tex-math id="M1">\begin{document}$ W^{1,p} $\end{document}</tex-math></inline-formula> estimates for composite material problems which can be described by a divergence form elliptic system on a nonsmooth domain composed of a finite number of subdomains. We want to derive global <inline-formula><tex-math id="M2">\begin{document}$ W^{1,p} $\end{document}</tex-math></inline-formula> regularity under the assumption that the coefficients are almost <inline-formula><tex-math id="M3">\begin{document}$ (\delta,R) $\end{document}</tex-math></inline-formula>-vanishing of codimension 1 (see Definition 1.2) in each of multiple subdomains and the boundaries of subdomains are Reifenberg flat, moreover the estimates do not depend on the distance between these subdomains.</p>