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arXiv · 2009.01050

The strong $L^p$-closure of vector fields with finitely many integer singularities on $B^3$

Riccardo Caniato
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Abstract

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 $p\ge 3/2$). 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.

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BibTeXRIS

Riccardo Caniato. 2020-09-02. The strong $L^p$-closure of vector fields with finitely many integer singularities on $B^3$. https://doi.org/10.1016/j.jfa.2021.109095

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