Dirichlet parabolicity and L1-Liouville property under localized geometric conditions

We shed a new light on the L1-Liouville property for positive, superharmonic functions by providing many evidences that its validity relies on geometric conditions localized on large enough portions of the space. We also present examples in any dimension showing that the L1-Liouville property is str...

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Bibliographic Details
Published inJournal of functional analysis Vol. 273; no. 2; pp. 652 - 693
Main Authors Pessoa, Leandro F., Pigola, Stefano, Setti, Alberto G.
Format Journal Article
LanguageEnglish
Published Elsevier Inc 15.07.2017
Subjects
[formula omitted]-Liouville property
Dirichlet parabolicity
Manifolds with boundary
L1-Liouville property
Dirichlet parabolicity
Manifolds with boundary
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Summary:We shed a new light on the L1-Liouville property for positive, superharmonic functions by providing many evidences that its validity relies on geometric conditions localized on large enough portions of the space. We also present examples in any dimension showing that the L1-Liouville property is strictly weaker than the stochastic completeness of the manifold. The main tool in our investigations is represented by the potential theory of a manifold with boundary subject to Dirichlet boundary conditions. The paper incorporates, under a unifying viewpoint, some old and new aspects of the theory, with a special emphasis on global maximum principles and on the role of the Dirichlet Green's kernel.
ISSN:0022-1236
1096-0783
DOI:10.1016/j.jfa.2017.03.016