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

Simão Antunes, Pedro Ricardo; Freitas, Pedro; Kennedy, James Bernard

ESAIM: Control, Optimisation and Calculus of Variations, Tome 19 (2013) no. 2, pp. 438-459.

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.

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