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 非线性椭圆型偏微分方程组的几个问题 二维和三维空间中的微分方程。 的 要考虑的具体系统来自理论， 非线性弹性，首席研究员将研究 作为能量最小化器的解的性质 相关的变分问题 她的主要目标是 确定解的各种正则性， 检查他们的定性行为。 使用的主要工具 是主要研究者关于 最大值原理和先验估计。 许多描述物理问题的解决方案 现实就是数学家所谓的“变分”解 也就是说，正确的物理解是数学解。 一个变分问题的解 在日常生活中，A 变分解是一个最大化或最小化 基本量，如系统的总能量。 为 例如，啤酒杯中的气泡是球形的，因为 球形使表面能最小化。 在这 项目主要研究人员将研究各种 变分问题解的数学性质， 模拟二维和三维非线性弹性现象。

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.