Certain min-max values related to the $p$-energy and packing radii of Riemannian manifolds and metric measure spaces

Abstract

Grosjean proved that the $(1/p)$-th power of the first eigenvalue of the $p$-Laplacian on a closed Riemannian manifold converges to the twice of the inverse of the diameter of the space, as $p \to \infty$. Before this, a corresponding result for the Dirichlet first eigenvalues was also obtained by Juutinen, Lindqvist and Manfredi. We extend those results for certain $k$-th min-max value related to the $p$-energy, where the corresponding limits are packing radii introduced by Grove-Markvorsen or its variant. Furthermore, we remark that our result holds for more singular setting.