Mathematical Sciences: Nonlinear Partial Differential Equations"

这个项目从两个不同的领域研究微分方程，与非线性弹性相关的变分演算和非线性系统的缺陷演化。对非线性弹性问题解的正则性的继续研究建立在最近建立的结果的基础上，这些结果表明二维弹性的某些边值问题具有局部Lipschitz连续同态的解。现在将努力推导这些解的估计，并利用它们来解决一般边值问题的正则性问题。还将研究一类特殊的椭圆变分问题，即聚凸问题，以确定这类问题解的梯度估计，并研究其解中存在孤立奇点的可能性。第二项工作涉及抛物线型金兹堡-朗道系统的光滑解。对于每一个固定的时刻，解被看作是空间域上的向量场。缺陷被定义为字段为空的地方。研究了缺陷图的演变及其对系统非线性结构的依赖。特别是对缺陷的创造现象、相互湮灭现象和稳定模式形成现象进行了研究。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创建方程，不如说是提供关于解的定性和定量信息。这可能包括回答关于独特性、平滑性和增长性的问题。此外，分析常常发展出解的近似方法和对这些近似精度的估计。＊＊＊

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. ***