Small perturbations of the type of boundary condition of an elliptic PDE and application to shape optimization
数学专题报告
报告题目(Title)：Small perturbations of the type of boundary condition of an elliptic PDE and application to shape optimization
报告人(Speaker)：Prof. Eric Bonnetier (Université Grenoble-Alpes)
地点(Place)：后主楼 1220
时间(Time)： 2025年 5 月 22 日（周四）10:00 am--11:00 am
邀请人(Inviter)：李海刚
报告摘要
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 $\omega_\varepsilon$ of the boundary. We characterize the first term in the asymptotic expansion of the solution, in terms of the relevant measure of smallness of $\omega_\varepsilon$, and we give explicit examples when $\omega_\varepsilon$ is a small surfacic ball in $ {\mathbb 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.
主讲人简介
Eric Bonnetier is a full professor at Université Grenoble-Alpes. He received his PhD in Applied Mathematics at the University of Maryland in 1988, under the guidance of Ivo Babuska. Eric Bonnetier has visited at Rutgers University, MSRI Berkeley, IMA University of Minnesota etc. His topics of interest include the mathematical modeling of composite materials, and particularly, the regularity of the fields in inhomogemeous media that contain close to touching inclusions. He is a coordinator of the Jean Kuntzmann Prize a distinguished lectures series of the Grenoble math/computer science community and a member of the editorial board of Computational and Applied Mathematics.
