Variational methods for Ginzburg-Landau systems

这项研究的目标是对物理学中出现的变分问题和相关的偏微分方程组(PDE)的解进行严格的数学分析。Ginzburg-Landau模型最初是在超导的背景下引入的，但类似的数学模型已经在物理系统的研究中变得无处不在，包括玻色-爱因斯坦凝聚、微磁体、共聚物和液晶。在某些极限状态下，可以观察到解发展出几何奇点，例如涡旋、倾斜或磁区壁，而这些缺陷给出了系统最显著的特征。 这个研究计划的总体目标是开发新的分析工具来研究奇异摄动的Ginzburg-Landau系统及其几何奇异性。所提出的问题不同于以前的工作，因为它们涉及向量值函数，产生非线性偏微分方程组。通常用于研究单个偏微分方程组的许多工具(如显式解、比较原理、相平面分析、刘维尔定理)不容易推广到系统。我为授权期提出的项目涉及具体的Ginzburg-Landau系统，它们表现出两种类型的奇点，旋涡和磁区壁，它们的解决将使人们对Ginzburg-Landau模型中奇点形成的本质有更深入的了解。解决这些问题将需要发展研究非线性偏微分方程组系统的新技术，将我自己的想法和方法与非线性和几何分析中各个领域的创新相结合。例如，这包括尖锐的能量界(通过涡球结构或类似的浓度测量)；单调性和ETA椭圆性方法(如在研究调和映射中开发的)；分叉技术；伽马收敛技术(用于识别表征奇点形状和相互作用的极限能量)；以及集中紧致性方法。这些数学上的进步部分是由物理洞察力和形式计算提出的，但将基于非线性分析方法和偏微分方程正则性理论。所获得的分析结果将使我们对这些模型及其所描述的现象有一个更全面、更可靠的理解，同时也为分析、几何和物理之间的丰富相互作用提供了新的视角。

The object of this research proposal is the rigorous mathematical analysis of variational problems arising in physics, and of the solutions of the associated systems of partial differential equations (PDE). The Ginzburg-Landau model was originally introduced in the context of superconductivity, but mathematical models of a similar kind have become ubiquitous in the study of physical systems, including Bose-Einstein condensation, micromagnets, copolymers, and liquid crystals. In certain limiting regimes the solutions are observed to develop geometrical singularities, such as vortices, disclinations, or domain walls, and these defects give the most salient features of the system. The overall goal in this research program is to develop new analytical tools to study singularly perturbed Ginzburg-Landau systems and their geometrical singularities. The proposed problems differ from previous work in that they concern vector-valued functions, yielding systems of nonlinear PDE. Many tools normally employed in studying a single PDE (such as explicit solutions, comparison principles, phase-plane analysis, Liouville theorems) do not extend easily to systems. The projects I propose for the grant period concern specific Ginzburg-Landau systems exhibiting singularities of two types, vortices and domain walls, and their resolution will yield insights into the nature of singularity formation in Ginzburg-Landau models in general. Solving them will entail the development of new techniques for studying systems of nonlinear PDE, by melding my own ideas and methods with innovations coming from various areas in nonlinear and geometric analysis. For example, this includes sharp energy bounds (via vortex-ball constructions or similar measures of concentration); monotonicity and eta-ellipticity methods (as developed in studying harmonic maps); bifurcation techniques; Gamma-convergence techniques (for identifying limiting energies which characterize singularity shape and interactions); and concentration-compactness methods. These mathematical advances will be suggested in part by physical insight and formal calculations, but will be based on methods of nonlinear analysis and PDE regularity theory. The analytical results obtained will give a more complete and reliable understanding of these models and the phenomena they describe, while providing new perspectives on the rich interplay between analysis, geometry, and physics.