The Poisson equation on Riemannian manifolds with weighted Poincaré inequality at infinity

We prove an existence result for the Poisson equation on non-compact Riemannian manifolds satisfying weighted Poincaré inequalities outside compact sets. Our result applies to a large class of manifolds including, for instance, all non-parabolic manifolds with minimal positive Green's function...

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Bibliographic Details
Published inarXiv.org
Main Authors Catino, Giovanni, Dario Daniele Monticelli, Punzo, Fabio
Format Paper
LanguageEnglish
Published Ithaca Cornell University Library, arXiv.org 02.05.2019
Subjects
Curvature
Green's functions
Infinity
Poisson equation
Riemann manifold
Theorems
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Summary:We prove an existence result for the Poisson equation on non-compact Riemannian manifolds satisfying weighted Poincaré inequalities outside compact sets. Our result applies to a large class of manifolds including, for instance, all non-parabolic manifolds with minimal positive Green's function vanishing at infinity. On the source function we assume a sharp pointwise decay depending on the weight appearing in the Poincaré inequality and on the behavior of the Ricci curvature at infinity. We do not require any curvature or spectral assumptions on the manifold.
ISSN:2331-8422