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

TL;DR

This paper characterizes the strong $L^p$-closure of vector fields with finitely many integer singularities in the unit ball, revealing how the structure changes with different integrability levels and providing a decomposition theorem.

Contribution

It provides a complete characterization of the strong $L^p$-closure for vector fields with integer singularities and introduces a decomposition theorem for these fields.

Findings

Characterization of the strong $L^p$-closure for all $p eq 3/2$

Identification of special behavior when $p o 3/2$

Decomposition theorem linking singularities to mass-minimizing currents

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.