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 $d\ge 3$, we prove that any field $f\in L^2(\partial \Omega ;{\mathbb{R}}^d)$ decomposes, in an unique way, as the sum of three silent 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 silent 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.

References in corpus (1)

• Hardy-Hodge Decomposition of Vector Fields in $Rn$