Asymptotic behaviour and numerical approximation of optimal eigenvalues of the Robin laplacian

Pedro Ricardo Simão Antunes; Pedro Freitas; James Bernard Kennedy

ESAIM: Control, Optimisation and Calculus of Variations (2013)

  • Volume: 19, Issue: 2, page 438-459
  • ISSN: 1292-8119

Abstract

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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.

How to cite

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Simão Antunes, Pedro Ricardo, Freitas, Pedro, and Kennedy, James Bernard. "Asymptotic behaviour and numerical approximation of optimal eigenvalues of the Robin laplacian." ESAIM: Control, Optimisation and Calculus of Variations 19.2 (2013): 438-459. <http://eudml.org/doc/272935>.

@article{SimãoAntunes2013,
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 &lt; α &lt; αn and, combined with analytic estimates, that this value is expected to grow with n1/N.},
author = {Simão Antunes, Pedro Ricardo, Freitas, Pedro, Kennedy, James Bernard},
journal = {ESAIM: Control, Optimisation and Calculus of Variations},
keywords = {Robin laplacian; eigenvalues; optimisation; Robin Laplacian},
language = {eng},
number = {2},
pages = {438-459},
publisher = {EDP-Sciences},
title = {Asymptotic behaviour and numerical approximation of optimal eigenvalues of the Robin laplacian},
url = {http://eudml.org/doc/272935},
volume = {19},
year = {2013},
}

TY - JOUR
AU - Simão Antunes, Pedro Ricardo
AU - Freitas, Pedro
AU - Kennedy, James Bernard
TI - Asymptotic behaviour and numerical approximation of optimal eigenvalues of the Robin laplacian
JO - ESAIM: Control, Optimisation and Calculus of Variations
PY - 2013
PB - EDP-Sciences
VL - 19
IS - 2
SP - 438
EP - 459
AB - 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 &lt; α &lt; αn and, combined with analytic estimates, that this value is expected to grow with n1/N.
LA - eng
KW - Robin laplacian; eigenvalues; optimisation; Robin Laplacian
UR - http://eudml.org/doc/272935
ER -

References

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