Remark on the Helmholtz decomposition in domains with noncompact boundary

TL;DR

This paper investigates the Helmholtz decomposition in unbounded domains with noncompact boundaries, demonstrating its validity in specific anisotropic spaces that include infinite energy vector fields.

Contribution

It shows that Helmholtz decomposition can hold in anisotropic spaces for domains with noncompact boundaries, extending known results beyond the energy space.

Findings

Helmholtz decomposition valid in certain anisotropic spaces

Includes some infinite energy vector fields

Applicable to domains with Lipschitz graph boundaries

Abstract

Let $\Omega$ be a domain with noncompact boundary. It is known that the Helmholtz decomposition is not always valid in $L^p(\Omega)$ except for the energy space $L^2 (\Omega)$. In this paper we consider a typical unbounded domain whose boundary is given as a Lipschitz graph, and show that the Helmholtz decomposition holds in certain anisotropic spaces which include some infinite energy vector fields.