Singular Poisson equations on Finsler-Hadamard manifolds

TL;DR

This paper investigates the properties of Sobolev spaces and establishes key results for singular Poisson equations involving the Finsler-Laplace operator on Finsler-Hadamard manifolds, focusing on reflexivity, uniqueness, and rigidity.

Contribution

It introduces new results on the reflexivity of Sobolev spaces and provides novel uniqueness and rigidity theorems for singular Poisson equations on Finsler-Hadamard manifolds.

Findings

Sobolev spaces are reflexive on certain Finsler manifolds

Uniqueness of solutions to singular Poisson equations is established

Rigidity results characterize solutions on Finsler-Hadamard manifolds

Abstract

In the first part of the paper we study the reflexivity of Sobolev spaces on non-compact and not necessarily reversible Finsler manifolds. Then, by using direct methods in the calculus of variations, we establish uniqueness, location and rigidity results for singular Poisson equations involving the Finsler-Laplace operator on Finsler-Hadamard manifolds having finite reversibility constant.