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Functions with prescribed singularities

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 Γ-convergence result for functionals of Ginzburg-Landau type, as described in [2]. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Journal of the European Mathematical Society Springer Journals

Functions with prescribed singularities

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Publisher
Springer Journals
Copyright
Copyright © 2003 by Springer-Verlag
Subject
Mathematics
ISSN
1435-9855
eISSN
1435-9863
DOI
10.1007/s10097-003-0053-5
Publisher site
See Article on Publisher Site

Abstract

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 Γ-convergence result for functionals of Ginzburg-Landau type, as described in [2].

Journal

Journal of the European Mathematical Society – Springer Journals

Published: Jul 10, 2003

References

  • A new proof of the rectifiable slices theorem
    Jerrard, R.L.