Noncompact variational problems involving complex unimodular maps


Petru Mironescu

09:10:00 - 10:00:00

101 , Mathematics Research Center Building (ori. New Math. Bldg.)

Among the complex-valued harmonic maps in the unit disc which are unimodular on the unit circle and vanish somewhere, Moebius transforms have minimal Dirichlet energy. This simple fact extends to more general energies and to multiply connected domains. The corresponding result is an essential tool in the analysis of several noncompact problems: uniqueness of vortexless minimizers for the Ginzburg-Landau equation, existence of degree one maps which are minimal in critical function spaces, existence of critical points of the Ginzburg-Landau energy with prescribed degree in simply connected domains. In a related direction, I will present a strategy for obtaining critical points of the Ginzburg-Landau energy with semi-stiff boundary conditions starting from Dirichlet boundary conditions. Based on joint works with L.V. Berlyand, A. Farina, X. Lamy, E. Sandier and V. Rybalko.

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