Functions with prescribed singularities

Abstract

The distributional k-dimensional Jacobian of a map u in the Sobolev space W1,k-1 which takes values in the the sphere Sk-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 Sk-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, as described in [2].