Uniqueness of vortexless Ginzburg-Landau type minimizers in two dimensions |
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Authors: | Alberto Farina Petru Mironescu |
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Institution: | 1. Faculté de Sciences, LAMFA, CNRS UMR 6140, Université de Picardie Jules Verne, 33, Rue Saint Leu, 80039, Amiens Cedex 1, France 2. Université de Lyon, Université Lyon 1, INSA de Lyon, Ecole Centrale de Lyon, UMR 5208, Institut Camille Jordan, 43 blvd du 11 novembre 1918, 69622, Villeurbanne Cédex, France
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Abstract: | In a simply connected two dimensional domain Ω, we consider Ginzburg-Landau minimizers u with zero degree Dirichlet boundary condition ${g \in H^{1/2}(\partial \Omega; \mathbb{S}^1)}$ . We prove uniqueness of u whenever either the energy or the Ginzburg-Landau parameter are small. This generalizes a result of Ye and Zhou requiring smoothness of g. We also obtain uniqueness when Ω is multiply connected and the degrees of the vortexless minimizer u are prescribed on the components of the boundary, generalizing a result of Golovaty and Berlyand for annular domains. The proofs rely on new global estimates connecting the variation of |u| to the Ginzburg-Landau energy of u. These estimates replace the usual global pointwise estimates satisfied by ${\nabla u}$ when g is smooth, and apply to fairly general potentials. In a related direction, we establish new uniqueness results for critical points of the Ginzburg-Landau energy. |
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