Remarks on Nodal Sets of Equations with Magnetic Schrodinger Operator
首发时间:2010-03-01
Abstract:In this paper we review some recent progress on the nucleation of zeros and the structure of the nodal sets of the complex-valued solutions of Ginzburg-Landau system for superconductivity and of more general partial differential equations with magnetic Schrodinger operator. These problems were motivated by the Ginzburg-Landau theory of superconductivity and by the Landau-de Gennes theory of liquid crystals. We review some results on the structure of the vortex sets of the solutions of Ginzburg-Landau system in the two dimensional case. The corresponding question in the three dimensional case has been less understood. Then we review some results on the nucleation of the vortices in the superconductors as the applied magnetic field increases. Finally we survey some recent progress on the structure of the singular sets of the complex-valued solutions of a general partial differential equation with a magnetic Schrodinger operator. Some unsolved problems are posed.
keywords: nodal set singular set superconductivity liquid crystals vortex vortex nucleation
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含有磁Schrodinger算子的偏微分方程复值解的零点集
摘要:本文考察了近年来对超导的Ginzburg-Landau方程复值解的零点的产生与零点集的结构的研究进展,也考察了含有磁Schrodinger算子的更一般的偏微分方程的相关问题。这些问题来源于超导的Ginzburg-Landau理论,与液晶的Landau-de Gennes 理论。我们考察了2维情形中Ginzburg-Landau方程的解的磁通涡漩集的结构的有关结果。在3维情形,对应的问题还没有深入研究。我们还考察了关于在外磁场增强时超导体中产生磁通涡漩的结果。最后我们总结了近年来关于含有磁Schrodinger算子的偏微分方程的复值解的临界集的结构的研究进展。本文提出了一些未解决的问题。
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No.4029451589112674****
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