We study a class of second-order elliptic equations and systems of divergence form, with discontinuous coefficients and data, which models the conductivity problem in composite materials. For the scalar case, we establish optimal gradient estimates by showing the explicit dependence of the elliptic coefficients and the distance between interfacial boundaries of inclusions. The novelty of these estimates is that they unify the known results in the literature and answer open problem (b) proposed by Li-Vogelius (2000) for the isotropic conductivity problem. We also obtain more interesting higher-order derivative estimates, which answers open problem (c) of Li-Vogelius (2000). This is based on a joint work with Hongjie Dong.

李海刚, 副教授。2007年国家建设高水平大学首批公派研究生, 北京师范大学 (导师：保继光教授) 与美国罗格斯 (Rutgers)大学(导师：李岩岩教授) 联合培养博士, 2009年6月获得博士学位。2009年8月留校工作至今, 2016年获霍英东教育基金会第十五届高等院校青年教师基金资助。他的主要研究兴趣是材料学和几何学中的线性和非线性偏微分方程理论。