Equivalences among parabolicity, comparison principle and capacity on complete Riemannian manifolds

TL;DR

This paper establishes new equivalences between p-parabolicity, comparison principles, and capacity on complete Riemannian manifolds, linking geometric properties with PDE behavior.

Contribution

It introduces a comparison principle for elliptic PDEs on exterior domains and proves its equivalence to p-parabolicity, extending known results to more general elliptic equations.

Findings

p-parabolicity is equivalent to the validity of the comparison principle for the p-Laplace equation

The comparison principle extends to more general elliptic PDEs on p-parabolic manifolds

Results can be adapted to non p-parabolic manifolds with growth conditions

Abstract

In this work we establish new equivalences for the concept of $p$-parabolic Riemannian manifolds. We define a concept of comparison principle for elliptic PDE's on exterior domains of a complete Riemannian manifold $M$ and prove that $M$ is $p$-parabolic if and only if this comparison principle holds for the $p$-Laplace equation. We show also that the $p$-parabolicity of $M$ implies the validity of this principle for more general elliptic PDS's and, in some cases, these results can be extended for non $p$-parabolic manifolds or unbounded solutions, provided that some growth of these solutions are assumed.