Uniqueness of critical points of the second Neumann eigenfunctions on triangles

Dear friends and colleagues,

in this Analysis and PDE seminar Associate Professor Ruofei Yao
from South China University of Technology is giving a talk on

Uniqueness of critical points of the second Neumann eigenfunctions on triangles

Abstract

The celebrated hot spots conjecture says that the second Neumann eigenfunctions
attain their (global) maximum (hottest point) only on the boundary of the domain.

Each vertex of a convex polygon in the plane is always a critical point of a Neumann
eigenfunction. In addition, Judge and Mondal [Ann. Math., 2022] showed that there
are no critical points in the interior of a triangle.

However, several open problems regarding the second Neumann eigenfunction in
triangles remained open.

In this talk, I answer some of these unresolved problems, including

    (1)    the uniqueness of non-vertex critical points,
    (2)    the sufficient and necessary conditions for the existence of a critical point,
   (3)    the exact location of the global maximum,
   (4)    the location of the nodal line,
   (5)    a new proof of the simplicity of the second Neumann eigenvalue,

and other. Our approach relies on the continuity method via domain deformation.

This is joint work with Prof. Hongbin Chen and Prof. Changfeng Gui.

Everyone interested in Analysis and PDEs is warmly invited to attend this seminar.

For our colleagues and friends in Armidale, you can join this seminar via Zoom:

    https://uni-sydney.zoom.us/j/81645174832

Best wishes,

Daniel H.