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 ${\left({\lambda }_i\right)}_{i=1}^{\infty }$ be the variational spectrum of ${\Delta }_p$ on a closed Riemannian manifold $(X,g)$ and let $N\left(\lambda \right)=\#\{i:{\lambda }_i<\lambda \}$ be the associated counting function. Then we have a Weyl law$N\left(\lambda \right)\sim c\mathrm{vol}\left(X\right){\lambda }^{n/p}.$ This confirms a conjecture of Friedlander. The proof is based on ideas of Gromov [5] and Liokumovich, Marques, Neves [7].