Asymptotic behaviour and numerical approximation of optimal eigenvalues of the Robin Laplacian^∗

¹ Group of Mathematical Physics of the University of Lisbon, Complexo Interdisciplinar, av. Prof. Gama Pinto 2, 1649-003 Lisboa, Portugal
pant@cii.fc.ul.pt; jkennedy@cii.fc.ul.pt
² Department of Mathematics, Universidade Lusófona de Humanidades e Tecnologias, av. do Campo Grande, 376, 1749-024 Lisboa, Portugal
³ Department of Mathematics, Faculty of Human Kinetics of the Technical University of Lisbon and Group of Mathematical Physics of the University of Lisbon, Complexo Interdisciplinar, av. Prof. Gama Pinto 2, 1649-003 Lisboa, Portugal
freitas@cii.fc.ul.pt
⁴ Institute of Applied Analysis, University of Ulm, Helmoltzstr. 18, 89069 Ulm, Germany

Received: 15 November 2011
Revised: 2 April 2012

Abstract

We consider the problem of minimising the nth-eigenvalue of the Robin Laplacian in R^N. Although for n = 1,2 and a positive boundary parameter α it is known that the minimisers do not depend on α, we demonstrate numerically that this will not always be the case and illustrate how the optimiser will depend on α. We derive a Wolf–Keller type result for this problem and show that optimal eigenvalues grow at most with n^1/N, which is in sharp contrast with the Weyl asymptotics for a fixed domain. We further show that the gap between consecutive eigenvalues does go to zero as n goes to infinity. Numerical results then support the conjecture that for each n there exists a positive value of α_n such that the nth eigenvalue is minimised by n disks for all 0 < α < α_n and, combined with analytic estimates, that this value is expected to grow with n^1/N.