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

ESAIM: Control, Optimisation and Calculus of Variations, 19 (2013), 438-459
http://dx.doi.org/10.1051/cocv/2012016

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 ? it is known that the minimisers do not depend on ?, we demonstrate numerically that this will not always be the case and illustrate how the optimiser will depend on ?. 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 ?n such that the nth eigenvalue is minimised by n disks for all 0 < ? < ?n and, combined with analytic estimates, that this value is expected to grow with n1/N.