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

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 ≤ 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.

Upper bounds for the Steklov eigenvalues of the $p$-Laplacian

“…Finally, we mention that a behavior similar to that of our case p > n has been observed for upper bounds on the Neumann eigenvalues of linear elliptic operators of order 2m, m ∈ N and density on Euclidean domains (see [8]) and for upper bounds on Neumann eigenvalues of the p-Laplacian in the conformal class of a given metric in a complete Riemannian manifold (see [9]).…”