ASYMPTOTIC BEHAVIOUR AND NUMERICAL APPROXIMATION OF OPTIMAL EIGENVALUES OF THE ROBIN LAPLACIAN

作者：Simao Antunes Pedro Ricardo; Freitas Pedro; Kennedy James Bernard

来源：ESAIM: Control, Optimisation and Calculus of Variations , 2013, 19(2): 438-459.

DOI：10.1051/cocv/2012016

• Robin Laplacian
• eigenvalues
• optimisation

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 a it is known that the minimisers do not depend on alpha, we demonstrate numerically that this will not always be the case and illustrate how the optimiser will depend on alpha. 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 alpha(n) such that the nth eigenvalue is minimised by n disks for all 0 %26lt; alpha %26lt; alpha(n) and, combined with analytic estimates, that this value is expected to grow with n(1/N)

• 出版日期2013-4