Dirichlet parabolicity and $L^1$-Liouville property under localized geometric conditions
Differential Geometry
2017-05-22 v2
Abstract
We shed a new light on the -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 -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.
Keywords
Cite
@article{arxiv.1607.06483,
title = {Dirichlet parabolicity and $L^1$-Liouville property under localized geometric conditions},
author = {Leandro F. Pessoa and Stefano Pigola and Alberto G. Setti},
journal= {arXiv preprint arXiv:1607.06483},
year = {2017}
}