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.

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.