In this paper we obtain essentially sharp generalized Keller–Osserman conditions for wide classes of differential inequalities of the form Lub(x)f(u)ℓ(|u|) and Lub(x)f(u)ℓ(|u|)−g(u)h(|u|) on weighted Riemannian manifolds, where L is a non-linear diffusion-type operator. Prototypical examples of these operators are the p-Laplacian and the mean curvature operator. The geometry of the underlying manifold is reflected, via bounds for the modified Bakry–Emery Ricci curvature, by growth conditions for the functions b and ℓ. A weak maximum principle which extends and improves previous results valid for the φ-Laplacian is also obtained. Geometric comparison results, valid even in the case of integral bounds for the modified Bakry–Emery Ricci tensor, are presented.

Keller-Osserman conditions for diffusion-type operators on Riemannian manifolds / L. Mari, M. Rigoli, A.G. Setti. - In: JOURNAL OF FUNCTIONAL ANALYSIS. - ISSN 0022-1236. - 258:2(2010), pp. 665-712. [10.1016/j.jfa.2009.10.008]

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

L. Mari
Primo
;
M. Rigoli
Secondo
;
2010

Abstract

In this paper we obtain essentially sharp generalized Keller–Osserman conditions for wide classes of differential inequalities of the form Lub(x)f(u)ℓ(|u|) and Lub(x)f(u)ℓ(|u|)−g(u)h(|u|) on weighted Riemannian manifolds, where L is a non-linear diffusion-type operator. Prototypical examples of these operators are the p-Laplacian and the mean curvature operator. The geometry of the underlying manifold is reflected, via bounds for the modified Bakry–Emery Ricci curvature, by growth conditions for the functions b and ℓ. A weak maximum principle which extends and improves previous results valid for the φ-Laplacian is also obtained. Geometric comparison results, valid even in the case of integral bounds for the modified Bakry–Emery Ricci tensor, are presented.
Keller–Osserman condition ; diffusion-type operators ; weak maximum principles ; weighted Riemannian manifolds ; quasi-linear elliptic inequalities
Settore MAT/03 - Geometria
Settore MAT/05 - Analisi Matematica
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Utilizza questo identificativo per citare o creare un link a questo documento: http://hdl.handle.net/2434/150477
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