Mathematical Sciences: Nonlinear Elasticity, Exterior Problems and Stochastic Control

该项目由四部分组成：非线性偏微分方程组、非线性弹性力学变分、随机控制和随机泛函微分方程。在非线性偏微分方程组部分，研究了拟线性椭圆型方程的两个特殊Dirichlet问题的径向对称解的存在性，其中一个问题含有平均曲率算子。他们希望使用变分法而不是基于常微分方程式的方法，得到与Serrin、Peletier和Ni相同的结果。在非线性弹性部分，作者提出了重新审视三维非线性弹性中的某些问题，从而产生了一维变分方法。他们建议利用鲍尔发现的弹性空化效应和拉夫伦蒂耶夫现象的高维形式之间的联系。在这两个随机部分中，研究人员希望研究具有规则或奇异控制的Bolza问题，并将这些思想应用于变分问题的另一类积分。他们计划在分析中加入记忆效应，并在随机泛函微分方程的分析中使用希尔伯特空间方法。这项研究工作属于应用分析和变分法，这是与弹性等物理问题接触的两个数学分支。这两位研究人员都是经验丰富的资深科学家，他们之前的数学成就功不可没。

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.