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Asymptotic behaviour and numerical approximation of optimaleigenvalues 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 RobinLaplacian in RN. Although for n = 1,2 and apositive boundary parameter α it is known that the minimisers do notdepend on α, we demonstrate numerically that this will not always be thecase and illustrate how the optimiser will depend on α. We derive aWolf–Keller type result for this problem and show that optimal eigenvalues grow at mostwith n1/N, which is in sharp contrast withthe Weyl asymptotics for a fixed domain. We further show that the gap between consecutiveeigenvalues does go to zero as n goes to infinity. Numerical results thensupport the conjecture that for each n there exists a positive value ofαn such that the ntheigenvalue is minimised by n disks for all0 < α < αnand, combined with analytic estimates, that this value is expected to grow withn1/N.

Keywords

Robin Laplacianeigenvaluesoptimisation

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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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