Conformal upper bounds for the eigenvalues of the p‐Laplacian

Bruno Colbois,Luigi Provenzano

doi:10.1112/jlms.12493

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 metric for 1 < p ⩽ n , and upper bounds for all p > 1 when we fix a metric. 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.

Highlights

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 Riemannian manifolds
As for Riemannian manifolds, we mention [12, 30, 32, 35, 41] for sharp estimates for the first positive eigenvalue in case of compact manifolds or domains and Neumann boundary conditions, under a given lower bound on the Ricci curvature
In this paper we investigate upper bounds for all variational eigenvalues that agree with the Weyl’s law under suitable geometrical assumptions on the manifold M

Summary

Introduction and statement of the main results

Let (M, g) be a complete, n-dimensional smooth Riemannian manifold, n ≥ 2, and let Ω ⊆ M be a bounded domain, i.e., a bounded connected open set, with boundary ∂Ω. As for Riemannian manifolds, we mention [12, 30, 32, 35, 41] for sharp estimates for the first positive eigenvalue in case of compact manifolds or domains and Neumann boundary conditions, under a given lower bound on the Ricci curvature. All in the same spirit of [3, 11, 21] have been proved for the Steklov problem [8], for hypersurfaces [9], and for the Neumann eigenvalues of the biharmonic operator on domains of Riemannian manifolds with Ricci curvature bounded from below [14]. We recall that Lemma 3.3 has been exploited to prove upper bounds for the eigenvalues on Σ, where Σ is an hypersurface in a complete n-dimensional Riemannian manifold (M, g) bounding some smooth domain Ω. In Appendix A we discuss upper bounds on hypersurfaces in terms of the isoperimetric ratio

Preliminaries and notation

Findings

Decomposition of a metric measure space by capacitors