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Functions with prescribed singularities

Giovanni Alberti; S. Baldo; G. Orlandi — 2003

Journal of the European Mathematical Society

The distributional $k$ -dimensional Jacobian of a map $u$ in the Sobolev space $W^{1, k - 1}$ which takes values in 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...

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