Mathematical Sciences: Nonlinear Partial Differential Equations"

9306199菲利普斯 本计画从两个不同的领域来研究微分方程，即与非线性弹性相关的变分法与非线性系统的缺陷演化。 继续工作的正则性的解决方案的问题，从非线性弹性建立在最近建立的结果表明，某些边界值问题的二维弹性的解决方案是局部Lipschitz连续同胚。 现在将努力得出这些解决方案的估计，并使用它们来解决一般边值问题的正则性问题。一类特殊的椭圆变分问题称为polyconvex也将进行研究，以确定这种类型的问题的解决方案的梯度估计，并调查孤立奇点在其解决方案的可能性。 第二行的工作涉及光滑的解决方案，抛物金斯堡-朗道系统。 对于每一个固定的时刻，解决方案被视为一个向量场在空间域。 缺陷被定义为字段为空的位置。 的缺陷模式的演变，以及它如何依赖于系统的非线性结构是要调查的。 特别是现象的创作，相互湮灭和稳定的图案形成的缺陷进行了研究。 偏微分方程是物理世界数学建模的基础。 数学分析的作用与其说是建立方程，不如说是提供关于解的定性和定量信息。 这可能包括关于独特性，平滑性和增长的问题的答案。 此外，分析经常发展出解的近似方法和对这些近似的准确性的估计。 ***

9306199 Phillips This project studies differential equations from two different areas, the calculus of variations related to nonlinear elasticity and the evolution of defects for nonlinear systems. Continuing work on the regularity of solutions to problems from nonlinear elasticity builds on recently established results showing that certain boundary value problems from two dimensional elasticity have solutions which are locally Lipschitz continuous homeomorphisms. Efforts will now be made to derive estimates for these solutions and to use them to address regularity questions for general boundary value problems. A particular class of elliptic variational problems called polyconvex will also be studied to determine gradient estimates for solutions to this type of problem and to investigate the possibility of isolated singularities in their solutions. The second line of work concerns smooth solutions to the parabolic Ginzburg-Landau system. For each fixed moment of time, the solution is viewed as a vector field over the spacial domain. A defect is defined as place where the field is null. The evolution of the defect pattern and how it depends on the nonlinear structure of the system is to be investigated. In particular the phenomena of creations, mutual annihilation and stable pattern formation of defects are to be studied. Partial differential equations form a basis for mathematical modeling of the physical world. The role of mathematical analysis is not so much to create the equations as it is to provide qualitative and quantitative information about the solutions. This may include answers to questions about uniqueness, smoothness and growth. In addition, analysis often develops methods for approximation of solutions and estimates on the accuracy of these approximations. ***