A Weyl law for the $p$-Laplacian

TL;DR

This paper establishes a Weyl law for the variational spectrum of the p-Laplacian on closed Riemannian manifolds, confirming a conjecture and extending classical spectral asymptotics to nonlinear operators.

Contribution

It proves a Weyl law for the p-Laplacian's spectrum, providing the first such result for this nonlinear operator and confirming Friedlander's conjecture.

Findings

Weyl law holds for the p-Laplacian spectrum

Asymptotic count of eigenvalues matches c * vol(X) * λ^{n/p}

Confirms Friedlander's conjecture for nonlinear spectral problems

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.