We study regularity and nonexistence properties for some free-interface problems arising in the study of limiting vorticities associated to the Ginzburg-Landau equations with magnetic field in two dimensions. Our results imply in particular that if these limiting vorticities concentrate on a smooth closed curve then they have a distinguished sign; moreover, if the domain is thin then solutions of the Ginzburg-Landau equations cannot have a number of vortices much larger than the applied magnetic field.
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Nam Q. Le, Regularity and nonexistence results for some free-interface problems related to Ginzburg–Landau vortices. Interfaces Free Bound. 11 (2009), no. 1, pp. 139–152DOI 10.4171/IFB/206