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.
 Publication:

arXiv eprints
 Pub Date:
 October 2019
 arXiv:
 arXiv:1910.11855
 Bibcode:
 2019arXiv191011855M
 Keywords:

 Mathematics  Spectral Theory;
 Mathematics  Analysis of PDEs;
 Mathematics  Differential Geometry;
 35P20;
 35P30