The aim of this work is twofold. The first main concern, the analytical one, is to study, using the method of gradient estimates, various Liouville-type theorems which are extensions of the classical Liouville Theorem for harmonic functions. We generalize the setting - from the Euclidean space to complete Riemannian manifolds - and the relevant operator - from the Laplacian to a general diffusion operator - and we also consider ``relaxed'' boundedness conditions (such as non-negativity, controlled growth and so on). The second main concern is geometrical, and is deeply related to the first: we prove some triviality results for Einstein warped products and quasi-Einstein manifolds studying a specific Poisson equation for a particular, and geometrically relevant, diffusion operator.

GRADIENT ESTIMATES AND LIOUVILLE THEOREMS FOR DIFFUSION-TYPE OPERATORS ON COMPLETE RIEMANNIAN MANIFOLDS / P. Mastrolia ; tutor: Marco Rigoli; coordinatore: Marco M. Peloso. Universita' degli Studi di Milano, 2011 Feb 11. 23. ciclo, Anno Accademico 2010. [10.13130/mastrolia-paolo_phd2011-02-11].

GRADIENT ESTIMATES AND LIOUVILLE THEOREMS FOR DIFFUSION-TYPE OPERATORS ON COMPLETE RIEMANNIAN MANIFOLDS

P. Mastrolia
2011

Abstract

The aim of this work is twofold. The first main concern, the analytical one, is to study, using the method of gradient estimates, various Liouville-type theorems which are extensions of the classical Liouville Theorem for harmonic functions. We generalize the setting - from the Euclidean space to complete Riemannian manifolds - and the relevant operator - from the Laplacian to a general diffusion operator - and we also consider ``relaxed'' boundedness conditions (such as non-negativity, controlled growth and so on). The second main concern is geometrical, and is deeply related to the first: we prove some triviality results for Einstein warped products and quasi-Einstein manifolds studying a specific Poisson equation for a particular, and geometrically relevant, diffusion operator.
11-feb-2011
Settore MAT/03 - Geometria
RIGOLI, MARCO
PELOSO, MARCO MARIA
Doctoral Thesis
GRADIENT ESTIMATES AND LIOUVILLE THEOREMS FOR DIFFUSION-TYPE OPERATORS ON COMPLETE RIEMANNIAN MANIFOLDS / P. Mastrolia ; tutor: Marco Rigoli; coordinatore: Marco M. Peloso. Universita' degli Studi di Milano, 2011 Feb 11. 23. ciclo, Anno Accademico 2010. [10.13130/mastrolia-paolo_phd2011-02-11].
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Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/2434/153097
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