Covariant Poisson equation with compact Lie algebras

TL;DR

This paper investigates the covariant Poisson equation for Lie algebra-valued functions in 3D space, establishing conditions for solutions' existence, smoothness, and asymptotic behavior using weighted covariant Sobolev spaces.

Contribution

It introduces weighted covariant Sobolev spaces and derives new existence and regularity conditions for solutions to the covariant Poisson equation.

Findings

Established sufficient conditions for solution existence and smoothness.

Derived weighted covariant Sobolev embedding theorems.

Analyzed asymptotic behavior of solutions.

Abstract

The covariant Poisson equation for Lie algebra-valued mappings defined in 3-dimensional Euclidean space is studied using functional analytic methods. Weighted covariant Sobolev spaces are defined and used to derive sufficient conditions for the existence and smoothness of solutions to the covariant Poisson equation. These conditions require, apart from suitable continuity, appropriate local integrability of the gauge potentials and global weighted integrability of the curvature form and the source. The possibility of nontrivial asymptotic behaviour of a solution is also considered. As a by-product, weighted covariant generalisations of Sobolev embeddings are established.