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 $N(\lambda) \sim c \operatorname{vol}(X) \lambda^{n/p}$. This confirms a conjecture of Friedlander. The proof is based on ideas of Gromov and Liokumovich, Marques, Neves.