Weak and strong $L^p$-limits of vector fields with finitely many integer singularities in dimension $n$

TL;DR

This paper characterizes the strong and weak $L^p$-limits of vector fields with finitely many integer singularities in various domains, providing new closure and characterization results.

Contribution

It identifies the strong $L^p$-closure of vector fields with integer singularities and proves weak sequential closure in certain domains, offering a new characterization via minimal connections.

Findings

Explicit description of strong $L^p$-closure $L_{ abla}^p(D)$.

Weak sequential closure of $L_{ abla}^p(D)$ in specific domains.

Characterization of such vector fields through minimal connection existence.

Abstract

For every given $p\in [1,+\infty)$ and $n\in\mathbb{N}$ with $n\ge 1$, the authors identify the strong $L^p$-closure $L_{\mathbb{Z}}^p(D)$ of the class of vector fields having finitely many integer topological singularities on a domain $D$ which is either bi-Lipschitz equivalent to the open unit $n$-dimensional cube or to the boundary of the unit $(n+1)$-dimensional cube. Moreover, for every $n\in\mathbb{N}$ with $n\ge 2$ the authors prove that $L_{\mathbb{Z}}^p(D)$ is weakly sequentially closed for every $p\in (1,+\infty)$ whenever $D$ is an open domain in $\mathbb{R}^n$ which is bi-Lipschitz equivalent to the open unit cube. As a byproduct of the previous analysis, a useful characterisation of such class of objects is obtained in terms of existence of a (minimal) connection for their singular set.