Uniqueness of vortexless Ginzburg-Landau type minimizers in two dimensions

To appear in Calc. Var. PDE, 28 p. DOI: 10.1007/s00526-012-0492-5International audienceIn 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