Mathematical Sciences: Harmonic Analysis and Partial Differential Equations

摘要Chanillo 9623079这个项目由十个问题组成，都是在偏微分方程领域提出的。其中八个问题与非线性偏微分方程组有关，两个问题是用来研究超定线性偏微分方程组的。非线性问题都是在物理和几何中出现的各种情况下产生的。该提案对偏微分方程组的关注比对单个方程的关注更强烈。之所以选择这些问题，部分是因为它们的解决方案和对它们的理解需要对经典的调和分析有所了解。物理系统的行为常常用偏微分方程式来描述。因此，知道如何解这些方程是很重要的。很多时候，有很多解决方案系列，因此人们可能真的不知道所有可能的解决方案。然而，如果通过某种技术，我们可以减少解决方案的列表，并表明只有特定类别的解决方案是唯一的可能性，那么确定所有解决方案的任务就变得更容易了。例如，如果一个方程表现出对称性，我们可以试着证明该方程的所有解都具有这种对称性。这将自动排除大类物体，并将我们的研究集中在一小类物体上，然后这类物体将告诉我们关于方程式的一切。最近，人们对一类被称为Ginzburg-Landau方程(GL方程)的方程的解产生了极大的兴趣。GL方程是以俄罗斯物理学家Ginzburg和Landau命名的，他们在20世纪50年代的S在超导理论中首次使用了GL方程。GL方程也出现在弦理论中。我们项目的主要部分致力于对Ginzburg-Landau方程的所有解进行分类。

Abstract Chanillo 9623079 The project consists of ten problems, all proposed in the field of Partial Differential Equations(PDE). Eight of the proposed problems pertain to questions arising in non-linear PDE, and two problems are proposed to study over-determined systems of linear partial differential equations. The non-linear problems all arise from various situations that arise in Physics and Geometry. There is a stronger focus in the proposal on systems of PDE than on single equations. The problems are selected in part because solutions to them and understanding them require some understanding of classical Harmonic analysis. Very often the behaviour of a physical system is described by partial differential equations. Thus it is important to know how to solve these equations. Very often there are many, many solution families and thus one may really not know all the possible solutions. However if by some technique we can cut down the list of solutions and show that only solutions of a certain category are the the only possibility then the task of determining all solutions become easier. For example if an equation exhibits symmetry one could try to show that all solutions of the equation will have this symmetry manifest. This will automatically rule out large classes of objects and focus our study on a small class which will then tell us everything about the equation. Recently there has been a great surge of interest in solutions of a type of equation called the Ginzburg-Landau equation (GL equations), named after the Russian physicists, Ginzburg and Landau who first used them in theories of superconductivity in the 1950's. The GL equations also appear in string theories. A major part of our project is devoted to classiying all the solutions of the Ginzburg-Landau equation.