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Journal Articles Calc. Var. Partial Differential Equations Year : 2013

Uniqueness of vortexless Ginzburg-Landau type minimizers in two dimensions

Abstract

In a simply connected two dimensional domain $\Omega$, 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 $\Omega$ 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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Dates and versions

hal-00747410 , version 1 (31-10-2012)

Identifiers

  • HAL Id : hal-00747410 , version 1

Cite

Alberto Farina, Petru Mironescu. Uniqueness of vortexless Ginzburg-Landau type minimizers in two dimensions. Calc. Var. Partial Differential Equations, 2013, 46 (3-4), pp.523--554. ⟨hal-00747410⟩
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