Higher regularity for solutions to equations arising from composite materials

We consider parabolic systems in divergence form with piecewise $C^{(s+\delta )\mathrm{/}2,s+\delta }$ coefficients and data in a bounded domain consisting of a finite number of cylindrical subdomains with interfacial boundaries in $C^{s+1+\mu }$, where $s\in \mathbb{N}$, $\delta \in (1\mathrm{/}2,1)$, and $\mu \in (0,1]$. We establish piecewise $C^{(s+1+{\mu }^{\mathrm{\prime }})\mathrm{/}2,s+1+{\mu }^{\mathrm{\prime }}}$ estimates for weak solutions to such parabolic systems, where ${\mu }^{\mathrm{\prime }}=\min \{1\mathrm{/}2,\mu \}$, and the estimates are independent of the distance between the interfaces. In the elliptic setting, our results answer an open problem © in Li and Vogelius (Arch. Rational Mech. Anal. 153 (2000), 91–151).