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In this paper, we discuss the global minimizers of a free energy for the superconducting thin films placed in a magnetic field hex below the lower critical field Hc1 or between Hc1 and the upper critical field Hc2. For hex is near but smaller than Hc1, we prove that the global minimizer having no vortex is unique. For Hc1 << hex << Hc2, we prove that the density of the vortices of the global minimizer is proportional to the applied field.

The global minimizers and vortex solutions to a Ginzburg-Landau model of superconducting films