The Helmholtz decomposition of a $BMO$ type vector field in general unbounded domains

TL;DR

This paper establishes a Helmholtz decomposition for a class of vector fields with bounded mean oscillation in arbitrary unbounded domains with smooth boundaries, extending classical results to more general settings.

Contribution

It introduces a Helmholtz decomposition for BMO-type vector fields in unbounded domains with smooth boundaries, broadening the scope of classical vector calculus results.

Findings

Helmholtz decomposition is valid for BMO vector fields in unbounded domains.

The decomposition holds in domains with uniformly $C^3$ boundaries for dimensions $n \,\geq\, 3$.

The study extends classical results to more general unbounded domains.

Abstract

We consider a space of $L^2$ vector fields with bounded mean oscillation whose ``normal'' component to the boundary is well-controlled. In the case when the dimension $n \geq 3$, we establish its Helmholtz decomposition for arbitrary uniformly $C^3$ domain in $\mathbf{R}^n$.