报告人：桂长峰教授（澳门大学）
报告地点：维格堂319
报告时间：5月26日下午2：30-3：30.
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 proposed by Judge and Mondal in [Ann. Math., 2020], but also accomplish a key objective outlined in the Polymath research thread led by Terence Tao. Furthermore, we resolve an eigenvalue inequality conjectured bySiudeja [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，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。桂长峰教授现致力于非线性偏微分方程的研究，特别是在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影响的工作，在Ann. of Math., Invent. Math., Comm. Pure Appl. Math.,Arch. Ration. Mech. Anal., Adv. Math.等国际顶级期刊上发表论文80余篇。
邀请人：王云