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Functions with prescribed singularities. (English) Zbl 1033.46028
The purpose of this paper is to investigate the range of the Jacobian operator, or, more precisely, to study which surfaces, or currents, can be obtained as Jacobians of \(S^{k-1}\)-valued maps.
The main result of the paper (Theorem 5.6 in the work) can be stated as follows. Let \(\Omega\) be an open set in \(\mathbb R^n\), and let \(M\) be the boundary of a rectifiable current of codimension \(k-1\) in \(\Omega\). Then there exists a map \(u\in W^{1,k-1}(\Omega, S^{k-1})\) whose Jacobian agrees, up to a canonical identification, with the current \(M\). The proof of this result is based on a dipole construction combined with an iteration argument.
In the case where \(M\) is polyhedral, the map constructed in the proof is smooth outside \(M\) plus an additional polyhedral set of lower dimension, and it can be used in the constructive part of the proof of a \(\Gamma\)-convergence result for functionals of Ginzburg-Landau type. If \(k=2\), the authors provide a simple proof of this result based on the theory of \(BV\) functions. The arguments given in the proofs are based on refined elliptic estimates combined with various approximation results for rectifiable currents.

MSC:
46E35 Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems
53C65 Integral geometry
49Q15 Geometric measure and integration theory, integral and normal currents in optimization
26B10 Implicit function theorems, Jacobians, transformations with several variables
58A25 Currents in global analysis
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