Mathematical Sciences: Harmonic Analysis and Partial Differential Equations

摘要Chanillo 9623079 该项目由十个问题组成，所有问题都是在 偏微分方程（PDE）其中八个问题涉及 提出了研究超定线性偏微分方程组的两个问题 方程非线性问题都是由各种情况引起的 在物理学和几何学中出现的。有一个更强的重点， 在偏微分方程组上比在单个方程上更有效。选择这些问题的部分原因是， 需要对经典谐波分析有所了解。 一个物理系统的行为通常是用部分的 微分方程因此，重要的是知道如何解决 这些方程式。通常有很多很多的解决方案 因此人们可能真的不知道所有可能的解决方案。 然而，如果我们可以通过某种技巧来削减解的列表，并表明只有某一类解才是最优解， 唯一的可能性，然后确定所有解决方案的任务成为 容易例如，如果一个方程表现出对称性，则可以尝试 证明了方程的所有解都具有这种对称性。 这将自动排除大类的对象，并集中我们的研究 一个小类，它会告诉我们关于这个方程的一切。 最近，人们对一种称为金兹伯格-朗道方程（GL方程）的方程的解产生了极大的兴趣， 以俄国物理学家金兹伯格和朗道的名字命名 在20世纪50年代将其用于超导理论。 GL方程也出现在弦理论中。的主要部分 我们的项目致力于分类所有的解决方案， Ginzburg-Landau方程

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.