The strong \(L^p\)-closure of vector fields with finitely many integer singularities on \(B^3\) (Q2039833)

The goal of this interesting paper is to investigate the set \(L^p_{\mathbb{Z}}(B)\) of the vector fields in the open unit ball \(B\subset\mathbb{R}^3\) which can be strongly \(L^p\)-approximated by vector fields that are smooth on \(B\) up a finite singular set \(\{x_1,\ldots, x_n\}\). The paper is organized into seven sections as follows : Introduction (useful notation and conventions, goal of the paper and related literature, statement of the main results), Basics on Sobolev principal \(U(1)\)-bundles over Lipschitz submanifolds in \(\mathbb{R}^n\), Choice of the cubic decompositions, Smoothing on the boundary of the cubic decompositions, Harmonic approximation on the good cubes, Radial approximation on the bad cubes, Final result and useful consequences (characterization of the class \(L^p_{\mathbb{Z}}(B)\) when \(p\geq \frac{3}{2}\), a decomposition theorem for vector fields in \(L^1_{\mathbb{Z}}(B)\)). Two useful appendices (Harmonic extensions of \(1\)-forms on open cubes, A useful characterization of divergence free vector fields in \(L^1_{\mathrm{loc}}(\Omega)\)) are given at the end of the paper.