Mathematical Sciences: "Variational Problems from Nonlinear Elasticity in Several Space Dimensions"

在这个项目中，首席研究员将研究几个涉及二维和三维空间中非线性椭圆型偏微分方程组的问题。要考虑的具体系统来自非线性弹性理论，首席研究员将研究相关变分问题的能量最小化解的性质。她的主要目标是确定解的各种规律性，并检查它们的定性行为。使用的主要工具是主要研究者关于最大原则和先验估计的先前结果。许多描述物理现实的问题的解都是数学家所说的“变分”解，也就是说，正确的物理解是变分问题的数学解。在普通语言中，变分解是使一个基本量（如系统的总能量）最大化或最小化的解。例如，啤酒中的气泡是球形的，因为球形使表面能最小化。在这个项目中，首席研究员将研究模拟二维和三维非线性弹性现象的变分问题解的各种数学性质。

In this project the principal investigator will examine several problems involving nonlinear elliptic systems of partial differential equations in two and three space dimensions. The specific systems to be considered come from the theory of nonlinear elasticity, and the principal investigator will study the properties of solutions that are the energy minimizers of the associated variational problems. Her primary goal is to determine various regularity properties of solutions and to examine their qualitative behavior. The main tools to be used are previous results of the principal investigator concerning maximum principles and a'priori estimates. The solutions of many problems that describe physical reality are what mathematicians call "variational" solutions, that is, the correct physical solution is the mathematical solution of a variational problem. In ordinary language, a variational solution is one that maximizes or minimizes a fundamental quantity like the total energy of a system. For example, the bubbles in your glass of beer are spherical, because the spherical shape minimizes the surface energy. In this project the principal investigator will study various mathematical properties of solutions of variational problems that model two- and three-dimensional nonlinear elastic phenomena.