We consider the problem of minimising the nth-eigenvalue of the RobinLaplacian in RN. Although for n = 1,2 and apositive boundary parameter α it is known that the minimisers do notdepend on α, we demonstrate numerically that this will not always be thecase and illustrate how the optimiser will depend on α. We derive aWolf–Keller type result for this problem and show that optimal eigenvalues grow at mostwith n1/N, which is in sharp contrast withthe Weyl asymptotics for a fixed domain. We further show that the gap between consecutiveeigenvalues does go to zero as n goes to infinity. Numerical results thensupport the conjecture that for each n there exists a positive value ofαn such that the ntheigenvalue is minimised by n disks for all0 < α < αnand, combined with analytic estimates, that this value is expected to grow withn1/N.
