A Weyl law for the $p$-Laplacian

Abstract

We show that a Weyl law holds for the variational spectrum of the $p$-Laplacian. More precisely, let $({\lambda }_i{)}_{i=1}^{\infty }$ be the variational spectrum of ${\Delta }_p$ on a closed Riemannian manifold $(X,g)$ and let $N(\lambda )=\#\{i:{\lambda }_i<\lambda \}$ be the associated counting function. Then we have a Weyl law $N(\lambda )\sim c\mathrm{vol}(X){\lambda }^{n/p}$. This confirms a conjecture of Friedlander. The proof is based on ideas of Gromov and Liokumovich, Marques, Neves.