Variational methods for Ginzburg-Landau systems

本研究计划的目的是对物理学中出现的变分问题以及相关偏微分方程（PDE）系统的解进行严格的数学分析。 金兹伯格-朗道模型最初是在超导性的背景下引入的，但类似的数学模型在物理系统的研究中已经无处不在，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，我们观察到解会产生几何奇异性，如涡旋、向错或畴壁，而这些缺陷是系统最显著的特征。本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金斯堡-朗道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，产生系统的非线性偏微分方程。 通常用于研究单个偏微分方程的许多工具（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。 我提出的项目资助期间关注具体的金斯堡-朗道系统表现出两种类型的奇点，旋涡和域壁，他们的决议将产生洞察金斯堡-朗道模型的奇点形成的性质一般。 解决这些问题将需要发展新的技术来研究系统的非线性偏微分方程，通过融合我自己的想法和方法与创新来自各个领域的非线性和几何分析。 例如，这包括尖锐的能量边界（通过涡球结构或类似的浓度测量）;单调性和η-椭圆率方法（如在研究调和映射中开发的）;分叉技术;伽马收敛技术（用于识别表征奇异形状和相互作用的极限能量）;和浓度紧致性方法。 这些数学上的进步将部分地由物理洞察力和正式计算提出，但将基于非线性分析和PDE正则性理论的方法。 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完整和可靠的理解，同时为分析、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.