北航数学论坛学术报告

--- 分析、偏微分方程与动力系统讨论班(2025春季第9讲)

Uniqueness of critical points of the second Neumann eigenfunctions on triangles

桂长峰 教授

(澳门大学)

时间与地点：2025年5月28日（周三下午）16：00-17：00

时间与地点：沙河主E803

摘要: 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, the 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.

报告人简介: 桂长峰，澳门大学数学系讲座教授，数学系主任，澳大发展基金会数学杰出学者，博士生导师。1991年在美国明尼苏达大学获博士学位。桂长峰教授曾入选国家级人才计划和海外高层次人才，于2013年当选美国数学会首届会士，获得过IEEE最佳论文奖、加拿大太平洋数学研究所研究成果奖、加拿大数学中心Andrew Aisensdadt奖等荣誉。他主要从事偏微分方程理论研究，特别在Allen-Cahn方程的研究、Moser-Trudinger不等式最佳常数的猜想、De Giorgi 猜想和Gibbons 猜想等方面取得了一系列在国际上有重大影响的工作，在国际一流数学学术期刊发表论文80余篇，其中包括Annals of Mathematics, Inventiones Mathematicae等顶级期刊。

邀请人：戴蔚 彭发

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