Gradient estimates for divergence form elliptic systems arising from composite material

In this paper, we show that $W^{1,p}$ $(1\le p<\mathrm{\infty })$ weak solutions to divergence form elliptic systems are Lipschitz and piecewise $C^1$ provided that the leading coefficients and data are of piecewise Dini mean oscillation, the lower order coefficients are bounded, and interfacial boundaries are $C^{1,\text{Dini}}$. This extends a result of Li and Nirenberg (\textit{Comm. Pure Appl. Math.} $\mathbf{\text{56}}$ (2003), 892-925). Moreover, under a stronger assumption on the piecewise $L^1$-mean oscillation of the leading coefficients, we derive a global weak type-(1,1) estimate with respect to $A_1$ Muckenhoupt weights for the elliptic systems without lower order terms.