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

Published online by Cambridge University Press:  16 January 2013

Pedro Ricardo Simão Antunes ,
Pedro Freitas  and
James Bernard Kennedy
Pedro Ricardo Simão Antunes
Affiliation:
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 Department of Mathematics, Universidade Lusófona de Humanidades e Tecnologias, av. do Campo Grande, 376, 1749-024 Lisboa, Portugal
Pedro Freitas
Affiliation:
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 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
James Bernard Kennedy
Affiliation:
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 Institute of Applied Analysis, University of Ulm, Helmoltzstr. 18, 89069 Ulm, Germany
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Abstract

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.

Keywords

Robin Laplacianeigenvaluesoptimisation
Type
Research Article
Information
ESAIM: Control, Optimisation and Calculus of Variations , Volume 19 , Issue 2 , April 2013 , pp. 438 - 459
Copyright
© EDP Sciences, SMAI, 2013

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