Remark on the Helmholtz decomposition in domains with noncompact boundary

Let $Ω$ be a domain with noncompact boundary. It is known that the Helmholtz decomposition is not always valid in $L^p(Ω)$ except for the energy space $L^2 (Ω)$. 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.

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