中科院数学与系统科学研究院
数学研究所
中科院华罗庚数学重点实验室
华罗庚青年数学论坛
综合报告
报告人： 李林涵 博士（University of Minnesota）
题 目：Elliptic Partial Differential Equations and Geometry
时 间：2022.04.22（星期五），09:00-10:00
地 点：数学院南楼N204室 腾讯会议：375-892-544
摘 要：There has been a great deal of interest in boundary value problems with minimal regularity assumptions on the coefficients of the equation, and/or on the boundary of the domain in question. The solvability of the Dirichlet problem for divergence-form elliptic operators with boundary data in $L^p$ turns out to have profound connections with many areas of analysis and geometric measure theory. In the first part of my talk, I’ll provide a survey on methods and results for the $L^p$ Dirichlet problem in Lipschitz domains. In the second part of my talk, I’ll talk about recent activities in understanding the precise relation between the solvability of the $L^p$ Dirichlet problem (a PDE property) and the geometric properties of the domain where the equation is given. Lastly, I’ll mention recent investigations into domains with higher co-dimensional boundaries (e.g. 3-dimensional space deprived of a curve) and how that motivates us to study some new PDE properties.
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