Decomposition of $L^{2}$-vector fields on Lipschitz surfaces: characterization via null-spaces of the scalar potential

TL;DR

This paper characterizes the decomposition of $L^{2}$-vector fields on Lipschitz surfaces into silent and Hardy components, revealing orthogonality conditions and explicit projections, especially for spherical boundaries.

Contribution

It introduces a unique decomposition of vector fields on Lipschitz surfaces into silent and Hardy parts, with explicit layer potential representations and orthogonality conditions.

Findings

Decomposition into silent vector fields is unique for Lipschitz surfaces.

Orthogonality of the decomposition occurs if and only if the surface is a sphere.

Explicit layer potential formulas for orthogonal projections are provided.

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 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.