The mathematical problem of gradient estimates for solutions of divergence form elliptic systems with piece-wise smooth coefficients arises in studying composite materials in applied science.

We will start with ideas in joint works with Vogelius (2000) and Nirenberg (2003) about a decade ago, in particular an open problem in the paper with Nirenberg, then discuss recent progress in closely related topics, such as gradient estimates for solutions of the Lame system with partially infinite coefficients (Arch. Rational Mech. Anal. (2015), joint with JiGuang Bao and HaiGang Li).

This is an expository lecture accessible to first year graduate students.