Conformal upper bounds for the eigenvalues of the p-Laplacian

In this note we present upper bounds for the variational eigenvalues of the (Formula presented.) -Laplacian on smooth domains of complete (Formula presented.) -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 (Formula presented.), and upper bounds for all (Formula presented.) 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.