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进行严格的数学分析，并对相关的偏微分方程组（PDE）的解进行分析。金兹堡-朗道模型最初是在超导的背景下引入的，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。在某些极限情况下，可以观察到解的几何奇点，如涡旋、偏斜或畴壁，这些缺陷给出了系统的最显著特征。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兹堡-朗道系统及其几何奇点。所提出的问题不同于以往的工作，因为它们涉及向量值函数，产生非线性偏微分方程系统。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）不容易扩展到系统中。我在拨款期间提出的项目涉及特定的金兹堡-朗道系统，它们表现出两种类型的奇点，涡旋和域壁，它们的分辨率将产生对金兹堡-朗道模型中奇点形成本质的见解。通过将我自己的想法和方法与来自非线性和几何分析各个领域的创新相结合，解决这些问题将需要开发研究非线性偏微分方程系统的新技术。例如，这包括尖锐的能量界限（通过涡球结构或类似的集中措施）；单调性和椭圆性方法（在研究谐波映射时发展起来的）；分岔技术;伽马收敛技术（用于识别奇点形状和相互作用特征的极限能量）；以及浓缩-压实法。这些数学上的进步将部分地由物理洞察力和形式计算提出，但将基于非线性分析和偏微分方程规则理论的方法。所获得的分析结果将对这些模型及其所描述的现象提供更完整、更可靠的理解，同时为分析、几何和物理之间丰富的相互作用提供新的视角。

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.