arXiv 2009.053372020-09-11math.AP

Lipschitz曲面上$L^{2}$-向量场的分解：标量势的零空间刻画

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

L. Baratchart, C. Gerhards, A. Kegeles

摘要：

对于$\partial Ω$是$\mathbb{R}^{d}$中有界连通强Lipschitz域的边界，本文证明了任何场$f\in L^{2} (\partial Ω; \mathbb{R}^{d})$以唯一的方式分解为三个无声的向量场之和，这些场的磁势在$\mathbb{R}^d\setminus\partial Ω$的一个或两个分量中为零.此外，该分解是正交的当且仅当$\partial Ω$是球面。我们还表明，任何$f$在$L^{2} (\partial Ω; \mathbb{R}^{d})$是唯一的两个沉默的字段和一个Hardy函数的总和，在这种情况下，无论$\partial Ω$的总和是正交的，我们表示相应的正交投影层的潜力。当$\partial Ω$是一个球，这两个分解重合，并在文献中被称为哈迪-霍奇分解匹配。

英文摘要：

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.

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摘要：

英文摘要：

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