Asymptotic Behaviour of Extremal Averages of Laplacian Eigenvalues

We study the convergence of extrema of averages of eigenvalues of the Dirichlet Laplacian on domains in (mathbb {R}^{n}) under both measure and surface measure restrictions. In the former case we prove that the sequence of averages to the power n / 2 is sub-additive and determine the first term in its asymptotics in the high-frequency limit. In the latter case, we show that the sequence of minimisers converges to the ball as the frequency goes to infinity. Similar results hold for Neumann boundary conditions.