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 approach to tackling the more than 50 year old conjecture of Pólya that the th eigenvalue of the Dirichlet Laplacian on any domain always lies above the corresponding first term in the Weyl asymptotics for that eigenvalue.
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.