Title: Dichotomy in gradient blow-up for the conductivity problems with imperfect bonding interfaces
Abstract: When two conductors are located close to each other, the electric field, represented by the gradient of solution to an elliptic PDE, may blow up as the distance between the inclusions approaches zero. This is called field concentration phenomenon, a central topic in composite material research. Classical results by Li-Vogelius and Li-Nirenberg reveal that there is no gradient blow-up when the conductivities of conductors are positive and finite. In this talk, I will discuss our work on a model involving imperfectly bonded conductors, where the transmission conditions are Robin-type boundary conditions. We proved a dichotomy in gradient blow-up driven by the bonding parameter. Moreover, at the critical value, we discovered a new blow-up phenomenon due to resonance, which disappears as the conductivities of conductors approaches infinity. This talk is based on joint work with Hongjie Dong (Brown U.) and Hanye Zhu (Duke U.).
Benjamin Weinkove
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