On higher regularity of Stokes systems with piecewise Hölder continuous coefficients

In this paper, we consider higher regularity of a weak solution $(\mathbf{u},p)$ to stationary Stokes systems with variable coefficients. Under the assumptions that coefficients and data are piecewise $C^{s,\delta }$ in a bounded domain consisting of a finite number of subdomains with interfacial boundaries in $C^{s+1,\mu }$, where $s$ is a positive integer, $\delta \in (0,1)$, and $\mu \in (0,1]$, we show that $D\mathbf{u}$ and $p$ are piecewise $C^{s,{\delta }_{\mu }}$, where ${\delta }_{\mu }=\min \{\frac{1}{2},\mu ,\delta \}$. Our result is new even in the 2D case with piecewise constant coefficients.