Functions with prescribed singularities |
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Authors: | G Alberti S Baldo and G Orlandi |
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Institution: | (1) Dipartimento di Matematica, Università di Pisa, via Buonarroti 2, 56127 Pisa, Italy;(2) Dipartimento di Matematica, Università di Trento, via Sommarive 14, 38050 Povo (Trento), Italy;(3) Dipartimento di Informatica, Università di Verona, Strada le Grazie 15, 37134 Verona, Italy |
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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 -convergence result for functionals of Ginzburg-Landau type, as described in 2]. Mathematics Subject Classification (2000) 46E35 (53C65, 49Q15, 26B10, 58A25) |
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Keywords: | Jacobian Sobolev maps singular maps integral currents rectifiability dipole construction complete intersections Brouwer degree coarea formula Ginzburg-Landau functionals |
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