In high-contrast composite materials, the stress concentration is a common phenomenon when inclusions are close to touch. It always causes damage initiation. This problem was proposed mathematically by Ivo Babuska, concerning the system of linear elasticity, modeled by a class of second order elliptic systems of divergence form with discontinuous coefficients.  I will first review some of our results on upper bound estimates by developing an iteration technique with respect to the energy integral to overcome the difficulty from the lack of maximal principle for elliptic systems. We real the relationship between the blow-up rate, the dimension, and the relative convexity. Finally, I will present two very recent results of myself on lower bound estimates and asymptotics of the gradients to show the optimality of the blow-up rates.

李海刚, 北京师范大学特聘副教授、博士生导师。北京师范大学与美国罗格斯(Rutgers)大学联合培养博士生, 2009年博士毕业, 留校工作至今。2016年获得教育部霍英东青年教师基金, 2017年获得教育部自然科学二等奖。主要研究复合材料中的Babuska问题, 针对高对比度的复合材料建立了Lame方程组解的梯度的最佳爆破估计, 揭示了纤维相对凸性在应力集中现象中的重要性, 改进了经典的De Giorgi-Nash-Morser理论在分片常系数椭圆方程情形的正则性结果。已在《Adv. Math.》(2篇)、《Arch. Ration. Mech. Anal.》 (2篇)、《Calc. Var. & PDEs》、《Trans. AMS》、《SIAM J. Math. Anal.》等主流数学杂志发表论文20余篇。