# A Weyl law for the $p$-Laplacian.

@article{Mazurowski2019AWL, title={A Weyl law for the \$p\$-Laplacian.}, author={Liam Mazurowski}, journal={arXiv: Spectral Theory}, year={2019} }

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.

## 5 Citations

### Upper bounds for the Steklov eigenvalues of the p‐Laplacian

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In this note, we present upper bounds for the variational eigenvalues of the Steklov p‐Laplacian on domains of Rn$\mathbb {R}^n$ , n⩾2$n\geqslant 2$ . We show that for 1n$p>n$ upper bounds depend on…

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### Certain min-max values related to the $p$-energy and packing radii of Riemannian manifolds and metric measure spaces

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Grosjean proved that the $(1/p)$-th power of the first eigenvalue of the $p$-Laplacian on a closed Riemannian manifold converges to the twice of the inverse of the diameter of the space, as $p \to…

### Parametric inequalities and Weyl law for the volume spectrum

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- 2022

We show that the Weyl law for the volume spectrum in a compact Riemannian manifold conjectured by Gromov can be derived from parametric generalizations of two famous inequalities: isoperimetric…

### Conformal upper bounds for the eigenvalues of the p ‐Laplacian

- MathematicsJournal of the London Mathematical Society
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In this note we present upper bounds for the variational eigenvalues of the p ‐Laplacian on smooth domains of complete n ‐dimensional Riemannian manifolds and Neumann boundary conditions, and on…

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