Uniqueness of vortexless Ginzburg-Landau type minimizers in two dimensions
Uniqueness of vortexless Ginzburg-Landau type minimizers in two dimensions
复制标题
二维无涡 Ginzburg-Landau 型最小化器的独特性
DOI：
--
复制
发表时间：
2013
期刊：
影响因子：
--
通讯作者：
P. Mironescu
中科院分区：
文献类型：
--
作者：
A. Farina;P. Mironescu
在单连通二维区域Ω中，我们考虑了具有零度Dirichlet边界条件的Ginzburg-Landau最小值u $${g in H^{1/2}(partial Omega; mathbb{S}^1)}$$。我们证明了当能量或金兹堡-朗道参数很小时u的唯一性。这推广了Ye和Zhou要求g光滑性的结果。当Ω是乘法连接并且在边界的分量上规定了无涡极小器u的度时，我们也得到了唯一性，推广了Golovaty和Berlyand关于环形区域的结果。这些证明依赖于将|和|的变化与u的金兹堡-朗道能量联系起来的新的全局估计。当g是光滑的时，这些估计取代了通常由$${
abla u}$$满足的全局点估计，并适用于相当一般的势。在相应的方向上，我们建立了新的金兹堡-朗道能量临界点的唯一性结果。
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 $${
abla u}$$ 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.