I will present some recent results on second-order divergence type equations with piecewise constant coefficients. This problem arises in the study of composite materials with closely spaced interface boundaries, and the classical elliptic regularity theory are not applicable. In the 2D case, we show that any weak solution is piecewise smooth without the restriction of the underling domain where the equation is satisfied. This completely answers a question raised by Li and Vogelius (2000) in the 2D case. Joint work with Hongjie Dong.