Ginzburg-Landau Vortices
Ginzburg-Landau Vortices
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DOI：
10.1142/5777
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发表时间：
1994
期刊：
影响因子：
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通讯作者：
F. Béthuel;H. Brezis;F. Hélein
中科院分区：
文献类型：
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作者：
F. Béthuel;H. Brezis;F. Hélein
本书中的数学直接适用于超导体、超流体和液晶的经典问题。从事现代材料研究的数学家、物理学家和工程师应该会对它感兴趣。本书涉及涉及小参数 E 的复值 Ginzburg-Landau 方程的二维稳态解 uE 的研究。此类问题与物理学中出现的问题有关，例如超导体和超流体中的相变现象。参数E具有长度的量纲，通常很小。因此，当 E 趋于零时，研究渐近应该会很有趣。主要结果之一断言最小化器 uE 的极限 u* 存在。此外，除了有限数量的点（物理学中称为缺陷或涡流）之外，u* 是平滑的。这些缺陷的数量恰好是边界条件的布劳威尔度（或缠绕数）。每个奇点都有一级——或者，正如物理学家所说，涡旋是量子化的。奇点具有无限的能量，但在去除核心能量后，我们得到了有限重正化能量的概念。奇点的位置完全是通过最小化所有可能的缺陷配置中的重整化能量来确定的。极限 u* 也可以被视为一个几何对象。它是具有规定边界条件 g 的 S1 的最小调和映射。拓扑障碍意味着每个映射 u 到 S1 且边界上 u=g 必须具有无限能量。尽管 u* 具有无限能量，但我们可以认为 u* 比边界上 u=g 的任何其他映射 u 具有“更少”的无限能量。本书中提供的材料大部分涵盖了作者最近的原创结果。它假设您对非线性泛函分析、偏微分方程和复杂函数有一定的了解。它是为研究人员和研究生设计的，可以作为一学期的教材使用。
The mathematics in this book apply directly to classical problems in superconductors, superfluids and liquid crystals. It should be of interest to mathematicians, physicists and engineers working on modern materials research. The text is concerned with the study in two dimensions of stationary solutions uE of a complex valued Ginzburg-Landau equation involving a small parameter E. Such problems are related to questions occuring in physics, such as phase transistion phenomena in superconductors and superfluids. The parameter E has a dimension of a length, which is usually small. Thus, it should be of interest to study the asymptotics as E tends to zero. One of the main results asserts that the limit u* of minimizers uE exists. Moreover, u* is smooth except at a finite number of points called defects or vortices in physics. The number of these defects is exactly the Brouwer degree - or winding number - of the boundary condition. Each singularity has degree one - or, as physicists would say, vortices are quantized. The singularities have infinite energy, but after removing the core energy we are led to a concept of finite renormalized energy. The location of the singularities is completely determined by minimizing the renormalized energy among all possible configurations of defects. The limit u* can also be viewed as a geometrical object. It is a minimizing harmonic map into S1 with prescribed boundary condition g. Topological obstructions imply that every map u into S1 with u=g on the boundary must have infinite energy. Even though u* has infinite energy one can think of u* as having "less" infinite energy than any other map u with u=g on the boundary. The material presented in this book covers mostly recent and original results by the authors. It assumes a moderate knowledge of nonlinear functional analysis, partial differential equations and complex functions. It is designed for researchers and graduate students alike and can be used as a one-semester text.