Gradient estimates for divergence form elliptic systems arising from composite material

TL;DR

This paper establishes gradient regularity results for divergence form elliptic systems with piecewise Dini mean oscillation coefficients, extending previous work and providing new global estimates under stronger conditions.

Contribution

It extends existing regularity results to systems with piecewise Dini mean oscillation coefficients and derives global weak type-(1,1) estimates under stronger assumptions.

Findings

Solutions are Lipschitz and piecewise $C^{1}$ under specified conditions.

Extended regularity results from Li and Nirenberg to more general coefficient conditions.

Derived global weak type-(1,1) estimates for systems without lower order terms.

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 $C^{1,\text{Dini}}$. This extends a result of Li and Nirenberg (\textit{Comm. Pure Appl. Math.} \textbf{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.