Geodesic
Functions with prescribed singularities
Journal of the European Mathematical Society, Tome 5 (2003) no. 3, pp. 275-311

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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].
DOI : 10.1007/s10097-003-0053-5
Classification : 46-XX, 26-XX, 49-XX, 53-XX
Keywords: Jacobian, Sobolev maps, singular maps, integral currents, rectifiability, dipole construction, complete intersections, Brouwer degree, coarea formula, Ginzburg-Landau functionals 
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     author = {Giovanni Alberti and S. Baldo and Giandomenico Orlandi},
     title = {Functions with prescribed singularities},
     journal = {Journal of the European Mathematical Society},
     pages = {275--311},
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Giovanni Alberti; S. Baldo; Giandomenico Orlandi. Functions with prescribed singularities. Journal of the European Mathematical Society, Tome 5 (2003) no. 3, pp. 275-311. doi: 10.1007/s10097-003-0053-5

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