Asymptotic expansions of the solution to an elliptic PDE in the presence of inclusions of small size has been a topic of great interest in the past decades. Indeed, such expansions proved interesting for applications in inverse problems, as a means to build efficient and stable algorithms for detecting inhomogeneities from boundary measurements. In this talk, we study the behavior of the solution to an elliptic equation when the boundary condition is perturbed on a small subset ω_ε of the boundary. We characterize the first term in the asymptotic expansion of the solution, in terms of the relevant measure of smallness of ω_ε, and we give explicit examples when ω_ε is a small surfacic ball in R^d,d=2,3.We use our asymptotic expansions to propose an algorithm for shape optimization problems, when the part of the boundary on which a specific boundary condition is prescribed is itself a design variable. This is joint work with Carlos Brito-Pacheco, Charles Dapogny, Rafael Estevez, and Michael Vogelius.
报告人简介：As a former student of Ecole Normale Supérieure (83-88)，Dr. Eric Bonnetier obtained his PhD in Applied Mathematics at the University of Maryland in 1988, under the enlightening guidance of Ivo Babuska. Dr. Bonnetier was then hired as chargé de recherche CNRS at the Center for Applied Maths of Ecole Polytechnique. He took a leave of absence from CNRS in 95-96. which I spent at Rutgers University. Since 2001, Dr. Bonnetier have been a full professor at Université Joseph Fourier in Grenoble (now part of Université GrenobleAlpes). After a sabbatical half-year at the MSRl in Berkeley in 2010,Dr.Bonnetier served as chairman of the Grenoble applied math departmen(Laboratoire Jean Kuntzmann) from 2011 to mid-2016. Dr. Bonnetier is currently at the Institut Fourier. His topics of interest include the mathematical modeling of composite materials, and particularly, the regularity of the fields (temperature，voltage potential, electromagnetic fields, elastic displacement...) in inhomogemeous media that contain close to touching inclusions. These questionshave strong connections to inverse problems, metamaterials, shape optimisat1orand fracture.
