Arkiv för Matematik

Volume 59 (2021)

Number 1

Optimal unions of scaled copies of domains and Pólya's conjecture

Pages: 11 – 51



Pedro Freitas (Departamento de Matemática, Instituto Superior Técnico, Universidade de Lisboa, Portugal; and Grupo de Física Matemática, Faculdade de Ciências, Universidade de Lisboa, Portugal)

Jean Lagacé (Department of Mathematics, University College London, United Kingdom)

Jordan Payette (Département de mathématiques et de statistique, Université de Montréal, Québec, Canada)


Given a bounded Euclidean domain $\Omega$, we consider the sequence of optimisers of the $k$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 Pólya’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 Pólya’s conjecture holds for a given specific domain or not. This approach allows us to recover a stronger version of Pólya’s original results for tiling domains satisfying some dynamical billiard conditions, and a strenghtening of Urakawa’s bound in terms of packing density.

Received 22 September 2020

Received revised 6 November 2020

Accepted 16 November 2020

Published 4 May 2021