Abstract: We consider the field concentration phenomenon in high-contrast composites, a central topic in composite material research. The governing equation is given by a divergence form elliptic equation. 

In this talk, I will first review recent breakthroughs regarding the insulated problem with perfect bonding. Then I will discuss our work on a model with imperfect bonding, where the potential function is allowed to be discontinuous across the interfacial boundaries. This model holds significant practical relevance, as it approximates the membrane structure in biological systems.

Based on joint work with Zhuolun Yang (Brown U.) and Hanye Zhu (Duke U.)

Bio Sketch: Hongjie Dong received his PhD in Mathematics from the University of Minnesota in 2005. He is currently a professor at the Division of Applied Mathematics of Brown University. His research lies in PDEs, probability theory, and numerical analysis; more specifically, he works on fully nonlinear elliptic and parabolic PDEs, probabilistic approach of PDEs, stochastic control theory, fluid equations, and more recently kinetic theory. He was awarded as a Simons Fellow in Mathematics in 2021, and won the NSF Career Award in 2011. He is on the editorial boards of many journals including SIAM J. Math. Anal. and J. Differential Equations.