Keller--Osserman conditions for diffusion-type operators on Riemannian Manifolds

TL;DR

This paper extends Keller-Osserman conditions to a broad class of nonlinear diffusion operators on Riemannian manifolds, linking geometric properties to solution non-existence criteria.

Contribution

It generalizes Keller-Osserman conditions for nonlinear diffusion operators on manifolds, incorporating geometric bounds and providing new non-existence results.

Findings

Conditions are necessary for non-existence of solutions.

Geometry influences growth conditions via Bakry-Emery Ricci curvature.

Extended maximum principle for inequalities involving these operators.

Abstract

In this paper we obtain generalized Keller-Osserman conditions for wide classes of differential inequalities on weighted Riemannian manifolds of the form $L u\geq b(x) f(u) \ell(|\nabla u|)$ and $L u\geq b(x) f(u) \ell(|\nabla u|) - g(u) h(|\nabla u|)$, where $L$ is a non-linear diffusion-type operator. Prototypical examples of these operators are the $p$-Laplacian and the mean curvature operator. While we concentrate on non-existence results, in many instances the conditions we describe are in fact necessary for non-existence. The geometry of the underlying manifold does not affect the form of the Keller-Osserman conditions, but is reflected, via bounds for the modified Bakry-Emery Ricci curvature, by growth conditions for the functions $b$ and $\ell$. We also describe a weak maximum principle related to inequalities of the above form which extends and improves previous results valid for the $\vp$-Laplacian.