时间：2021年8月2日-2021年8月7日 8:30--21:30

Time：8:30-21:30, August 2, 2021 - August 7, 2021

地点：西湖大学云栖校区4号楼409-411教室

Venue: Room 409-411, Building 4, Yunqi Campus, Westlake University

主持人：西湖大学理学院 赵永强 博士

Host: Dr. Yongqiang Zhao, School of Science, Westlake University

日程/Schedule:

1、主讲嘉宾/Speaker：Prof. Daguang Chen, Tsinghua University

主讲题目/Title：Eigenvalue estimates for Laplacian II

摘要/Abstract:

This short course will focus on the eigenvalue estimates on Riemannian manifolds. We will introduce Cheeger-Yau's heat kernel comparison theorem, S.Y. Cheng's eigenvalue estimates, Li-Yau's estimates, Zhong-Yang's inequality, Polya conjecture and universal inequalities for eigenvalues of Laplacian.

Reference：

R. Schoen and S. T. Yau, Lectures on differential geometry, International Press, 1994.

2、主讲嘉宾/Speaker：Prof. Hanlong Fang, Peking University

主讲题目/Title：Proof of the fundamental gap conjecture I

摘要/Abstract:

Let Ω⊂Rⁿ be convex and bounded. Denote by 0<λ₀<λ₁ the first two eigenvalues of the operator -Δ+V on Ω with zero Dirichlet boundary conditions, where we assume the potential V is weakly convex. Denote by D=sup_x,y∈Ω||y-x|| the diameter of Ω. Andrews-Clutterbuck proved that λ₁-λ₀>3π²/D². We will discuss their proof in detail.

主讲题目/Title： Proof of the fundamental gap conjecture II

摘要/Abstract:

We will continue the study of Andrews-Clutterbuck's proof on the fundamental gap conjecture.

主讲题目/Title：Stability estimates for the lowest eigenvalue of a Schrödinger operator

摘要/Abstract:

We will study the work of Carlen-Frank-Lieb on the stability of the lowest eigenvalue of the Schrödinger operator -Δ+V in L²(R^d) under the constraint of a given L^p norm of the potential.

3、主讲嘉宾/Speaker：Prof. Bing Xie, Shandong University

主讲题目/Title：Spectral gap of 1-dimensional Schrodinger problem with Robin boundary conditions

摘要/Abstract:

We will discuss an article by Mark S. Ashbaugh and Derek Kielty. In the paper, the authors prove sharp lower bounds on the spectral gap of 1-dimensional Schrodinger operators with Robin boundary conditions for each value of the Robin parameter. In particular, the lower bounds apply to single-well potentials with a centered transition point. This result extends work of Cheng et al. and Horvath in the Neumann and Dirichlet endpoint cases to the interpolating regime. The authors also build on recent work by Andrews, Clutterbuck, and Hauer in the case of convex and symmetric single-well potentials. In particular, the authors show the spectral gap is an increasing function of the Robin parameter for symmetric potentials.

4、主讲嘉宾/Speaker： Dr. Jiashu Zhang, Westlake University

主讲题目/Title：Maximization of the second non-trivial Neumann eigenvalue I

摘要/Abstract:

Let Ω∈Rⁿ be a bounded regular open set and μ₂ be the second nontrivial. Neumann eigenvalue of the Laplace operator on Ω. Denote B* a ball with volume |B| = |Ω|/2 and μ*₁ the first Neumann nontrivial eigenvalue of B*. We prove that  μ₂ ≤ μ*₁, where equality holds only if Ω coincides with the union of two disjoint equal balls almost everywhere.

主讲题目/Title：Maximization of the second non-trivial Neumann eigenvalue II

摘要/Abstract:

We will continue to prove that maximization of the second non-trivial Neumann eigenvalue is given by the union of two disjoint equal ball.

讲座联系人/Contact:

理学院 王老师  wangqiuhui@westlake.edu.cn