Functions with prescribed singularities

Alberti, Giovanni, Baldo, S., and Orlandi, G.. "Functions with prescribed singularities." Journal of the European Mathematical Society 005.3 (2003): 275-311. <http://eudml.org/doc/277650>.

@article{Alberti2003,
abstract = {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 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].},
author = {Alberti, Giovanni, Baldo, S., Orlandi, G.},
journal = {Journal of the European Mathematical Society},
keywords = {Jacobian; Sobolev maps; singular maps; integral currents; rectifiability; dipole construction; complete intersections; Brouwer degree; coarea formula; Ginzburg-Landau functionals; Ginzburg-Landau functional; Jacobian; Sobolev map; integral currents; Brouwer degree; dipole construction; coarea formula},
language = {eng},
number = {3},
pages = {275-311},
publisher = {European Mathematical Society Publishing House},
title = {Functions with prescribed singularities},
url = {http://eudml.org/doc/277650},
volume = {005},
year = {2003},
}

TY - JOUR
AU - Alberti, Giovanni
AU - Baldo, S.
AU - Orlandi, G.
TI - Functions with prescribed singularities
JO - Journal of the European Mathematical Society
PY - 2003
PB - European Mathematical Society Publishing House
VL - 005
IS - 3
SP - 275
EP - 311
AB - 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 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].
LA - eng
KW - Jacobian; Sobolev maps; singular maps; integral currents; rectifiability; dipole construction; complete intersections; Brouwer degree; coarea formula; Ginzburg-Landau functionals; Ginzburg-Landau functional; Jacobian; Sobolev map; integral currents; Brouwer degree; dipole construction; coarea formula
UR - http://eudml.org/doc/277650
ER -