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 suggests that the ball should in a certain sense be
asymptotically optimal.
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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 P\'olya 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.
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In a second part, we will present some recent analogous results for the
Laplacian with Robin boundary conditions, which are joint work with Pedro
Freitas.