There is an obvious topological obstruction for a finite energy unimodular harmonic extension of a S 1-valued function defined on the boundary of a bounded regular domain of R n. When such extensions do not exist, we use the Ginzburg-Landau relaxation procedure. We prove that, up to a subsequence, a sequence of Ginzburg-Landau minimizers, as the coupling parameter tends to infinity, converges to a unimodular harmonic map away from a codimension-2 minimal current minimizing the area within the homology class induced from the S 1-valued boundary data. The union of this harmonic map and the minimal current is the natural generalization of the harmonic extension.
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Received December 3, 1998 / final version received May 10, 1999
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Lin, F., Rivière, T. Complex Ginzburg-Landau equations in high dimensions and codimension two area minimizing currents. J. Eur. Math. Soc. 1, 237–311 (1999). https://doi.org/10.1007/s100970050008
- Mathematics Subject Classification (1991): 35J20, 35J25, 35J60, 35J50, 35Qxx, 49Q05, 49Q15, 49Q20, 53Z05, 58E15, 58E20, 81T13