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.

Dirichlet parabolicity and $L^1$-Liouville property under localized geometric conditions

PIGOLA, STEFANO;SETTI, ALBERTO GIULIO
2017-01-01

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.
http://www.sciencedirect.com/science/article/pii/S0022123617301416
L1-Liouville property Dirichlet parabolicity Manifolds with boundary
Pessoa, Leandro F.; Pigola, Stefano; Setti, ALBERTO GIULIO
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Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/11383/2062371
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