SIAM Journal on Mathematical Analysis, Volume 53, Issue 4, Page 4096-4117, January 2021.
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.

中文翻译：

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Lipschitz 表面上 $L^{2}$-矢量场的分解：通过标量势的零空间表征

SIAM 数学分析杂志，第 53 卷，第 4 期，第 4096-4117 页，2021 年 1 月。
对于 $\partial\Omega$ 在 $\mathbb{R}^{d}$ 与 $d\geq3$ 中的有界且连通的强 Lipschitz 域的边界，我们证明了任何域 $f\in L^{2} (\partial\Omega ; \mathbb{R}^{d})$ 以一种独特的方式分解为三个不可见矢量场的总和——其磁势在 $\mathbb{ 的一个或两个分量中消失的场R}^d\setminus\partial\Omega$。此外，这种分解是正交的当且仅当 $\partial\Omega$ 是一个球体。我们还表明 $L^{2}(\partial\Omega ; \mathbb{R}^{d})$ 中的任何 $f$ 都是两个不可见场和一个哈代函数的唯一和，在这种情况下，和不管 $\partial\Omega$ 是正交的；我们用层电位表示相应的正交投影。当 $\partial\Omega$ 是一个球体时，