The Helmholtz decomposition of a space of vector fields with bounded mean oscillation in a bounded domain

TL;DR

This paper introduces a new space of vector fields with bounded mean oscillation in bounded domains, establishing a Helmholtz decomposition that extends previous results from half-spaces to bounded domains.

Contribution

It extends the Helmholtz decomposition to a new space of vector fields with bounded mean oscillation in bounded domains, with distinct boundary component behaviors.

Findings

Helmholtz decomposition is valid in the new vector field space.

The result generalizes earlier half-space findings to bounded domains.

The space captures boundary component behaviors separately.

Abstract

We introduce a space of vector fields with bounded mean oscillation whose ``tangential'' and ``normal'' components to the boundary behave differently. We establish its Helmholtz decomposition when the domain is bounded. This substantially extends the authors' earlier result for a half space.