Conformal upper bounds for the eigenvalues of the $p$-Laplacian

TL;DR

This paper establishes conformal upper bounds for the eigenvalues of the $p$-Laplacian on various Riemannian manifolds, using a metric approach to construct test functions, aligning with known asymptotic estimates.

Contribution

It introduces new conformal upper bounds for $p$-Laplacian eigenvalues on Riemannian manifolds, extending previous results and providing bounds in terms of isoperimetric ratios.

Findings

Upper bounds match Friedlander's asymptotic estimates.

Bounds are provided in conformal classes and fixed metrics.

Results apply to manifolds with boundary and hypersurfaces.

Abstract

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 compact (boundaryless) Riemannian manifolds. In particular, we provide upper bounds in the conformal class of a given manifold $(M,g)$ for $1<p\leq n$, and upper bounds for all $p>1$ when we fix a metric $g$. To do so, we use a metric approach for the construction of suitable test functions for the variational characterization of the eigenvalues. The upper bounds agree with the well-known asymptotic estimate of the eigenvalues due to Friedlander. We also present upper bounds for the variational eigenvalues on hypersurfaces bounding smooth domains in a Riemannian manifold in terms of the isoperimetric ratio.