Gradient Estimates for Divergence Form Elliptic Systems Arising from Composite Material

Source abstract

In this paper, we show that $W^{1,p}$ $(1\leq p < \infty)$ 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 $C^{1,{Dini}}$. This extends a result of Li and Nirenberg [ Comm. Pure Appl. Math., 56 (2003), pp. 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.