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

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.