Sums of Dirichlet eigenvalues in proximity of the disk and the ball

Abstract

We derive second order perturbation expansions of volume-normalized sums of the first $n$ Dirichlet eigenvalues for near-circular and near-spherical domains. The disk (resp. ball) is only a critical shape when $n$ is the last index of an eigenvalue cluster, and in dimension two and three our expansions yield a finitely checkable criterion deciding whether the disk or ball is a weak local minimizer or a saddle point. In two dimensions, this allows us to show that there is an infinite number of cases in which the disk is a saddle point for finite eigenvalue sums.