The Helmholtz decomposition of a $BMO$ type vector field in a slightly perturbed half space

TL;DR

This paper develops a Helmholtz decomposition for a specific class of vector fields with bounded mean oscillation in a slightly perturbed half space, extending classical results to more general domains.

Contribution

It introduces a new space of vector fields with controlled boundary normal components and proves Helmholtz decomposition in perturbed half spaces.

Findings

Helmholtz decomposition holds in perturbed $C^3$ half spaces.

The space of vector fields with bounded mean oscillation is well-defined and suitable for decomposition.

Results extend classical Helmholtz theory to perturbed domains.

Abstract

We introduce a space of $L^2$ vector fields with bounded mean oscillation whose normal component to the boundary is well-controlled. We establish its Helmholtz decomposition in the case when the domain is a perturbed $C^3$ half space in $\mathbf{R}^n$ $(n \geq 3)$ with small perturbation.