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.