Optimal unions of scaled copies of domains and P\'olya's conjecture

Abstract

Given a bounded Euclidean domain $\Omega$, we consider the sequence of optimisers of the $k^{\rm th}$ Laplacian eigenvalue within the family consisting of all possible disjoint unions of scaled copies of $\Omega$ with fixed total volume. We show that this sequence encodes information yielding conditions for $\Omega$ to satisfy P\'{o}lya's conjecture with either Dirichlet or Neumann boundary conditions. This is an extension of a result by Colbois and El Soufi which applies only to the case where the family of domains consists of all bounded domains. Furthermore, we fully classify the different possible behaviours for such sequences, depending on whether P\'{o}lya's conjecture holds for a given specific domain or not. This approach allows us to recover a stronger version of P\'{o}lya's original results for tiling domains satisfying some dynamical billiard conditions, and a strenghtening of Urakawa's bound in terms of packing density.