Asymptotic behaviour and numerical approximation of optimal eigenvalues of the Robin Laplacian∗
Group of Mathematical Physics of the University of Lisbon,
Complexo Interdisciplinar, av.
Prof. Gama Pinto 2, 1649-003
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
4 Institute of Applied Analysis, University of Ulm, Helmoltzstr. 18, 89069 Ulm, Germany
Revised: 2 April 2012
We consider the problem of minimising the nth-eigenvalue of the Robin Laplacian in RN. 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 n1/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 αn such that the nth eigenvalue is minimised by n disks for all 0 < α < αn and, combined with analytic estimates, that this value is expected to grow with n1/N.
Mathematics Subject Classification: 35P15 / 35J05 / 49Q10 / 65N25
Key words: Robin Laplacian / eigenvalues / optimisation
P.R.S.A. was supported by Fundação para a Ciência e Tecnologia (FCT), Portugal, through grant SFRH/BPD/47595/2008 and project PTDC/MAT/105475/2008 and by Fundação Calouste Gulbenkian through program Estímulo à Investigação 2009. J.B.K. was partially supported by a grant within the scope of FCT’s project PTDC/MAT/101007/2008 and a fellowship of the Alexander von Humboldt Foundation, Germany. All authors were partially supported by FCT’s projects PTDC/MAT/101007/2008 and PEst-OE/MAT/UI0208/2011.
© EDP Sciences, SMAI, 2013