The strong Lp-closure of vector fields with finitely many integer singularities on B3

Abstract

This paper is aimed to investigate the strong Lp-closure LZp(B) of the vector fields on the open unit ball B⊂R3 that are smooth up to finitely many integer point singularities. First, such strong closure is characterized for arbitrary p∈[1,+∞). 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 LZ1(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.