PDE Seminar Abstracts

Uniqueness of critical points of the second Neumann eigenfunctions on triangles

Ruofei Yao
South China University of Technology, China
Mon 31st Jul 2023, 2-3pm, Carslaw Room 829 (AGR)


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.