Decomposition of $L^{2}$-Vector Fields on Lipschitz Surfaces: Characterization via Null-Spaces of the Scalar Potential

Abstract

For $\partial\Omega$ the boundary of a bounded and connected strongly Lipschitz domain in $\mathbb{R}^{d}$ with $d\geq3$, we prove that any field $f\in L^{2}(\partial\Omega ; \mathbb{R}^{d})$ decomposes, in a unique way, as the sum of three invisible vector fields---fields whose magnetic potential vanishes in one or both components of $\mathbb{R}^d\setminus\partial\Omega$. Moreover, this decomposition is orthogonal if and only if $\partial\Omega$ is a sphere. We also show that any $f$ in $L^{2}(\partial\Omega ; \mathbb{R}^{d})$ is uniquely the sum of two invisible fields and a Hardy function, in which case the sum is orthogonal regardless of $\partial\Omega$; we express the corresponding orthogonal projections in terms of layer potentials. When $\partial\Omega$ is a sphere, both decompositions coincide and match what has been called the Hardy--Hodge decomposition in the literature.