This paper is aimed to investigate the strong $L^p$-closure $L_{\mathbb{Z}}^p(B)$ of the vector fields on the open unit ball $B\subset {\mathbb{R}}^3$ that are smooth up to finitely many integer point singularities. First, such strong closure is characterized for arbitrary $p\in [1,+\infty )$. Secondly, it is shown what happens if the integrability order p is large enough (namely, if $p⩾3/2$). Eventually, a decomposition theorem for elements in $L_{\mathbb{Z}}^1(B)$ is given, conveying information about the possibility of connecting the singular set of such vector fields by a mass-minimizing, integer 1-current on B with finite mass.

中文翻译：

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B ³上具有有限许多整数奇点的矢量场的强L ^p闭包

本文旨在调查强 ${\mathrm{大号}}^p$-关闭 ${\mathrm{大号}}_{\mathbb{ž}}^p（乙）$ 单位球上矢量场的分布 $乙\subset {\mathbb{[R}}^3$平滑到有限的整数整数奇点。首先，这种强力的封闭具有任意性的特点$p\in [\mathrm{1个}，+\infty ）$。其次，显示出如果可积阶数p足够大（即，如果$p⩾3/\mathrm{2个}$）。最终，分解定理为${\mathrm{大号}}_{\mathbb{ž}}^{\mathrm{1个}}（乙）$给出了传递有关通过矢量以有限质量在B上进行质量最小化的整数1-电流连接这种矢量场的奇异集的可能性的信息。