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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| Published in | Journal of functional analysis Vol. 273; no. 2; pp. 652 - 693 |
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| Main Authors | , , |
| Format | Journal Article |
| Language | English |
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Elsevier Inc
15.07.2017
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| Abstract | 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. |
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| AbstractList | 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. |
| Author | Pessoa, Leandro F. Setti, Alberto G. Pigola, Stefano |
| Author_xml | – sequence: 1 givenname: Leandro F. surname: Pessoa fullname: Pessoa, Leandro F. email: leandropessoa@ufpi.edu.br organization: Departamento de Matemática, Universidade Federal do Piauí-UFPI, 64049-550, Teresina, Brazil – sequence: 2 givenname: Stefano surname: Pigola fullname: Pigola, Stefano email: stefano.pigola@uninsubria.it organization: Dipartimento di Scienza e Alta Tecnologia, Università dell'Insubria – Como, via Valleggio 11, 22100 Como, Italy – sequence: 3 givenname: Alberto G. surname: Setti fullname: Setti, Alberto G. email: alberto.setti@uninsubria.it organization: Dipartimento di Scienza e Alta Tecnologia, Università dell'Insubria – Como, via Valleggio 11, 22100 Como, Italy |
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| Title | Dirichlet parabolicity and L1-Liouville property under localized geometric conditions |
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