Topic: On Hotspots Conjecture for Domain with N-Axes of Symmetry

Lecturer: Prof. Yi Li

Time: September 3th, 2025, 9:30-12:30, UTC+8

Venue: Room 213, School of Mathematics and Statistics, Nanhu Campus

Biography: Professor Yi Li was elected a Fellow of the American Association for the Advancement of Science (AAAS Fellow) in 2018. He received his Bachelor’s degree in Mathematics from Xi’an Jiaotong University in 1982 and his Ph.D. from the University of Minnesota in 1988. Prof. Li’s research focuses primarily on nonlinear problems and their applications, spanning fields such as Physics, Geometry, and Biomedicine. He has long been dedicated to the study of differential geometry in astrophysics and the conformal curvature equation of the Matukuma equation. His research also involves bioluminescence tomography, and he co-invented a method for computational optical biopsy with G. Wang. Notably, he resolved a problem concerning the uniqueness and asymptotic behavior of solutions to semi-degenerate nonlinear equations posed by the renowned mathematicians Berestycki and Nirenberg. Additionally, together with collaborators, he solved a long-standing open problem regarding the stability of traveling-wave solutions of the nonlinear reaction-diffusion equations. Professor Yi Li has published over a hundred academic papers in internationally renowned mathematics journals like Arch. Rational Mech. Anal., Duke Math. J., JDE, Commu. PDE, CPAM., JPMA. He has served as the Academic Vice President and Provost at the City University of New York’s John Jay College of Criminal Justice, and previously held positions as Dean of the College of Science and Mathematics at Wright State University, and as Professor and Chair of the Department of Mathematics at the University of Iowa.

Abstract: In this talk, we prove the hot spots conjecture for rotationally symmetric domains in Rn by the continuity method. More precisely, we show that the odd Neumann eigenfunction in xn associated with lowest nonzero eigenvalue is a Morse function on the boundary, which has exactly two critical points and is monotone in the direction from its minimum point to its maximum point. As a consequence, we prove that the Jerison and Nadirashvili’s conjecture 8.3 holds true for rotationally symmetric domains and are also able to obtain a sharp lower bound for the Neumann eigenvalue. We will also discuss some recent results on n-axes symmetry or hyperbolic drum type domains.

Rewritten by: Mei Mengqi

Edited by: Li Tiantian

Source: School of Mathematics and Statistics