数学研究所

数学科学全国重点实验室

学术报告

偏微分方程研讨班

Speaker: 桂长峰教授 （澳门大学）

Inviter: 张平 院士

Language: Chinese

Title: Uniqueness of critical points of the second Neumann eigenfunctions on triangles

Time&Venue: 2025年5月21日（星期三）15:00-16:00& 南楼N820

Abstract: The hot spots conjecture, proposed by Rauch in 1974, asserts that the second Neumann eigenfunction of the Laplacian achieves its global maximum (the hottest point)

exclusively on the boundary of the domain. Notably, for triangular domains, the11 absence of interior critical points was recently established by Judge and Mondal in [Ann.

Math., 2022]. Nevertheless, several important questions about the second Neumann eigenfunction in triangles remain open. In this talk, we address issues such as: (1) the uniqueness of non-vertex critical points; (2) the necessary and sufficient conditions for the existence of non-vertex critical points; (3) the precise location of the global extrema;

(4) the position of the nodal line; among others. Our results not only confirm both the original theorem and Conjecture 13.6 proposed by Judge and Mondal in [Ann. Math.,2020], but also accomplish a key objective outlined in the Polymath 7 research thread 1 led by Terence Tao. Furthermore, we resolve an eigenvalue inequality conjectured by Siudeja [Proc. Amer. Math. Soc., 2016] concerning the ordering of mixed Dirichlet–Neumann Laplacian eigenvalues for triangles. Our approach employs the continuity method via domain deformation.This is a joint work with Hongbin Chen and Ruofei Yao.