Mathematical Sciences: Nonlinear Elasticity, Exterior Problems and Stochastic Control

该项目由四部分组成：非线性部分 微分方程，非线性变分法 弹性力学，随机控制与随机泛函 微分方程 在非线性偏微分方程部分， 研究人员希望研究是否存在径向对称的 两个特殊拟线性Dirichlet问题的解 椭圆方程，其中一个包含平均曲率 操作符.他们希望获得与塞林相同的结果， Peletier和Ni，使用变分法，而不是 方法基于常微分方程。 在非线性弹性部分，作者建议重新- 研究三维非线性弹性中的某些问题， 这就产生了一维变分方法。他们 建议使用弹性空化之间的连接 球发现的效果和一个更高的维度形式的 拉夫连季耶夫现象。 在研究者想要研究的两个随机部分中 具有规则或奇异控制的Bolza问题及其应用 这些想法的另一类被积函数的变分 问题他们计划将记忆效应融入他们的 分析和使用希尔伯特空间方法的分析， 随机泛函微分方程 本研究工作属于应用分析和微积分 这两个数学分支 比如弹性问题。两名研究人员都是 资深的经验丰富的科学家，他们的许多荣誉， 以前的数学成就。

The project consists of four parts: nonlinear partial differential equations, calculus of variations in nonlinear elasticity, stochastic control and stochastic functional differential equations. In the nonlinear partial differential equations part the investigators want to study the existence of radially symmetric solutions of two special Dirichlet problems for quasilinear elliptic equations, one of which contains a mean curvature operator. They would like to obtain the same results as Serrin, Peletier and Ni, using the calculus of variations rather than methods based on ordinary differential equations. In the nonlinear elasticity part the authors propose to re- examine certain issues in 3-dimensional nonlinear elasticity, which give rise to the one dimensional variational methods. They propose to use the connection between the elastic cavitation effect discovered by Ball and a higher dimensional form of the Lavrentiev's phenomenon. In the two stochastic parts the investigators want to study Bolza problems with regular or singular control, and to apply these ideas to another class of integrands for variational problems. They plan to incorporate memory effects in their analysis and use Hilbert space methods in the analysis of stochastic functional differential equations. This research effort belongs to applied analysis and calculus of variations, two branches of mathematics that come in contact with physical problems such as elasticity. Both investigators are senior experienced scientists with a lot of credit for their previous mathematical accomplishments.