Mathematical Physics and Harmonic Analysis Seminar

Date: February 9, 2018

Time: 1:50PM - 2:50PM

Location: BLOC 628

Speaker: James Kennedy, University of Lisbon

Title: Asymptotically optimal Laplacian eigenvalues and Polya's conjecture

Abstract: A longstanding problem in spectral geometry is to determine the domain(s) which minimise a given eigenvalue of a differential operator such as the Laplacian with Dirichlet boundary conditions, among all domains of given volume. For example, the Theorem of (Rayleigh--) Faber--Krahn states that the smallest eigenvalue is minimal when the domain is a ball. Very little to nothing is known about domains minimising the higher eigenvalues, but the Weyl asymptotics suggest that the ball should in a certain sense be asymptotically optimal.

In the first part of this talk, we will sketch a new approach to this problem initiated by a paper of Colbois and El Soufi in 2014, which asks not after the minimising domains themselves but properties of the corresponding sequence of minimal values. This serendipitously also yields a new Ansatz for tackling the more than 50 year old conjecture of Polya that the k-th eigenvalue of the Dirichlet Laplacian on any domain always lies above the corresponding first term in the Weyl asymptotics for that eigenvalue. Along the way, we will additionally meet variants of Gauss circle problem.

In a second part, we will present some recent analogous results for the Laplacian with Robin boundary conditions, which are ongoing joint work with Pedro Freitas.