Mathematical Analysis and Applications of Concentration Phenomena in Nonlinear Elliptic Equations

我的建议的主要目标是研究非线性椭圆型方程和系统，特别强调具有集中现象的解的问题。根据非线性的性质，有三种主要的集中现象：尖峰、过渡层和涡旋。尖峰出现在非线性薛定谔方程(NLS)、数学生物学中的反应扩散系统等中。过渡层与材料科学中的Allen-Cahn方程和相变有关。磁金兹堡-朗道方程、Chern-Simons-Higgs系统和Yang-Mills-Higgs系统模拟了涡旋在超导和粒子物理中的重要作用。这项提议包括两个部分。在第一部分(纯数学部分)，我们想要了解模拟集中现象的半线性椭圆型偏微分方程解的结构。这一部分的一个主要方面是将微分几何的思想引入到三个重要方程的整体解的分析和构造中：Allen-Cahn方程、非线性薛定谔方程和磁金兹堡-朗道方程。这些方程是半线性椭圆问题的典型代表。其目的是在一些标量方程的整体解的研究与极小曲面、常平均曲率(CMC)曲面和Toda系统的理论之间建立复杂的对应关系。在第二部分(应用数学部分)，我们计划研究如何将非线性偏微分方程组、科学计算、匹配渐近、变分方法、临界点理论、动力系统、微分几何和代数几何中的不同技术和结果应用于求解物理世界中的非线性方程。这些问题包括两嵌段共聚物理论中的界面行为，以及由反应扩散系统模拟的局部图案形成问题，以及在生物形态发生、理论化学、城市犯罪热点模式等方面的应用。

The primary goal of my proposal is to study nonlinear elliptic equations and systems with particular emphasis on problems with solutions that exhibit concentration phenomena. There are three main types of concentration phenomena depending on the nature of the nonlinearity: spikes, transition layers, and vortices. Spikes arise in Nonlinear Schrodinger equations (NLS), reaction diffusion sytem in mathematical biology, etc. Transition layers are associated with Allen-Cahn equation and phase trasitions in material sciences. Vortices play an important role in Superconductivity and Particle Physics modeled by the magnetic Ginzburg-Landau equation, Chern-Simons-Higgs system and Yang-Mills-Higgs system. There are two parts of this proposal. In the first part (pure mathematics part), we would like to understand the structure of entire solutions to semilinear elliptic PDEs modeling concentration phenomena. A major aspect of this part is to bring ideas from Differential Geometry into the analysis and construction of entire solutions for three important equations: the Allen-Cahn equation, the nonlinear Schrodinger equation and magnetic Ginzburg-Landau equation. These equations are typical representatives of semilinear elliptic problems. The objective is to establish an intricate correspondence between the study of entire solutions of some scalar equations and the theories of minimal surfaces, constant mean curvature (CMC) surfaces and Toda systems. In the second part (the applied mathematics part), we plan to investigate how different techniques and results in nonlinear PDEs, scientific computing, matched asymptotics, variational methods, critical point theory, dynamical systems, differential geometry and algebraic geometry can be applied to solve nonlinear equations arising from the physical world. Such problems include interface behavior in diblock copolymer theory, and localized pattern formation problems modeled by reaction-diffusion systems, with applications to biological morphogenesis, theoretical chemistry, hot-spot patterns of urban crime, etc.