Functions with prescribed singularities

The distributional $k$-dimensional Jacobian of a map $u$ in the Sobolev space $W^{1,k-1}$ which takes values in the the sphere $S^{k-1}$ can be viewed as the boundary of a rectifiable current of codimension $k$ carried by (part of) the singularity of $u$ which is topologically relevant. The main purpose of this paper is to investigate the range of the Jacobian operator; in particular, we show that any boundary $M$ of codimension $k$ can be realized as Jacobian of a Sobolev map valued in $S^{k-1}$. In case $M$ is polyhedral, the map we construct is smooth outside $M$ plus an additional polyhedral set of lower dimension, and can be used in the constructive part of the proof of a $\Gamma$-convergence result for functionals of Ginzburg-Landau type described in the paper "Variational convergence for functionals of Ginzburg-Landau type" by the same authors