Mathematical Analysis and Applications of Concentration Phenomena in Nonlinear Elliptic Equations

非线性椭圆方程集中现象的数学分析及应用

• 批准号: 435557-2013
• 负责人: Wei, Juncheng
• 金额: $ 2.77万
• 依托单位: University of British Columbia
• 依托单位国家: 加拿大
• 项目类别: Discovery Grants Program - Individual
• 财政年份: 2017
• 资助国家: 加拿大
• 起止时间: 2017-01-01 至 2018-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=635323
• 关键词: Mathematical; Analysis; Applications; Concentration; Phenomena

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.

我建议的主要目标是研究非线性椭圆方程和系统，特别着重于表现出浓度现象的解决方案问题。浓度现象的主要类型取决于非线性的性质：尖峰，过渡层和涡流。在非线性Schrodinger方程（NLS），数学生物学中的反应扩散系统等中出现峰值。过渡层与材料科学中的Allen-Cahn方程和相体发生相关。涡旋在由磁性金兹堡 - 兰道方程，Chern-Simons-Higgs系统和Yang-Mills-higgs系统建模的超导性和粒子物理学中起着重要作用。该提案有两个部分。在第一部分（纯数学部分）中，我们想了解半线性椭圆形PDES建模浓度现象的整个解决方案的结构。该部分的一个主要方面是将差异几何形状的思想带入三个重要方程的整个解决方案的分析和构建：allen-cahn方程，非线性schrodinger方程和磁性金兹堡 - 兰道方程。这些方程是半线性椭圆问题的典型代表。目的是在某些标量方程的整个解和最小表面的理论，恒定平均曲率（CMC）表面和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）中，我们计划研究非线性PDE，科学计算，匹配的渐近方法，变异方法，临界点理论，动力学系统，差异几何形状和代数几何形状如何应用于求解非线性方程来从物理世界中应用。此类问题包括二嵌段共聚物理论中的界面行为，以及由反应扩散系统建模的局部模式形成问题，以及用于生物形态发生的应用，理论化学，城市犯罪的热点模式，等等。