Title: The first Dirichlet eigenvalue and the width

Speaker：Xu Guoyi（Tsinghua University）

Time：March，20th，10：00-11：00

Place: Fifth Teaching Building, Classroom 5307

Abstract: There are a lot of result about the sharp lower bound of the first Dirichlet eigenvalue or Neumann eigenvalue under different restriction (back to Faber-Krahn and Payne-Weinberger for Euclidean domain, Li-Yau and Zhong-Yang for manifolds case). Generally, the sharp lower bounds of those eigenvalues are achieved on disk (sphere, for the case that boundary is empty, corresponding non-collapsed case) or line segment (circle, corresponding collapsed case). Recent years, there are research to characterize the difference between the domain and the model space (mentioned above), by the gap between the eigenvalue and its sharp bound (quantitative Faber-Krahn inequality etc). And the results along this direction obtained so far, in the spirit, are close to the quantitative isoperimetric inequality established during the last decade (Fusco-Maggi-Pratelli etc). The common point is that the model space is “homogenous in any direction (disk or sphere etc) and is non-collapsed. 

In this talk, we present our recent result, which gives the explicitly quantitative inequality, linking the width of the domain with the gap between the first Dirichlet eigenvalue and its sharp lower bound. One novel thing is that our model space is collapsed line segment. Most part of this talk only requires the basic PDE and Riemannian geometry knowledge.