Inserted: 23 jan 2003
Last Updated: 6 mar 2007
Journal: J. Eur. Math. Soc.
Final version, April 2003
The distributional k-dimensional Jacobian of a Sobolev map u which takes values in the (k-1)-dimensional sphere can be viewed as a rectifiable current of codimension k located on (a 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. We show that any boundary M of codimension k can be realized as Jacobian of a Sobolev map; and in case M is polyhedral, the map we construct is smooth outside M plus an additional polyhedral set of lower dimension.
Keywords: Rectifiability, coarea formula, distributional jacobian, integral currents, singular maps