论文标题
$ l^{2} $ - lipschitz表面上的向量字段的分解：通过标量电位的空空间进行表征
Decomposition of $L^{2}$-vector fields on Lipschitz surfaces: characterization via null-spaces of the scalar potential
论文作者
论文摘要
对于$ \部分ω$，带有$ d \ geq3 $的$ \ mathbb {r}^{d} $在$ \ mathbb {r \ geq3 $的边界的边界字段---在$ \ mathbb {r}^d \ setMinus \ partialω$的一个或两个组件中的磁电势消失的字段。此外，当且仅当$ \ partialω$是一个球体时，这种分解是正交的。我们还表明，任何$ f $ in $ l^{2}（\ partialω; \ mathbb {r}^{d}）$都是两个静音字段和一个强壮的函数的唯一总和，在这种情况下，总和是正交的，无论$ \ partialω$;我们从层势方面表达了相应的正交投影。当$ \partialΩ$是一个球体时，两个分解都重合并匹配文献中所谓的Hardy-Hodge分解。
For $\partial Ω$ the boundary of a bounded and connected strongly Lipschitz domain in $\mathbb{R}^{d}$ with $d\geq3$, we prove that any field $f\in L^{2} (\partial Ω; \mathbb{R}^{d})$ decomposes, in an unique way, as the sum of three silent vector fields---fields whose magnetic potential vanishes in one or both components of $\mathbb{R}^d\setminus\partial Ω$. Moreover, this decomposition is orthogonal if and only if $\partial Ω$ is a sphere. We also show that any $f$ in $L^{2} (\partial Ω; \mathbb{R}^{d})$ is uniquely the sum of two silent fields and a Hardy function, in which case the sum is orthogonal regardless of $\partial Ω$; we express the corresponding orthogonal projections in terms of layer potentials. When $\partial Ω$ is a sphere, both decompositions coincide and match what has been called the Hardy-Hodge decomposition in the literature.
