Mathematical Sciences: Nonlinear PDEs and Viscosity Solutions: Variational Problems and Control Theory

9300966延森这个项目将研究由变分和控制理论引起的数学问题。这些问题的一个共同特征是它们对粘度溶液框架的亲和力。解决这些问题所需的新技术和方法的发展既将推动粘度溶液理论的发展，也将为其适用性提供新的范例。1981年，为了唯一刻画最优控制和微分对策的值函数，人们引入了微分方程的粘性解。该项目的一部分将继续研究这类应用。另外两个问题领域来自变分演算。第一种是基于在具有极小L-1范数的区域上寻找具有给定边值的函数的策略。利用粘性溶液成功地研究了这些问题。目前的工作将使用粘性解来分析运动问题的平均曲率的具体解，以帮助理解这些函数的水平集。第二条线是研究与以前一样的函数集，目标是最小化最大范数。这就产生了四阶齐次偏微分方程组。对这些问题的分析已经做了一些工作，但还有更多的工作要做。偏微分方程式是建立物理世界数学模型的基础。数学分析的作用与其说是创建方程，不如说是提供有关解的定性和定量信息。这可能包括回答有关唯一性、平稳性和成长性的问题。此外，分析经常开发出近似解的方法和对这些近似的精度的估计。***

9300966 Jensen This project will examine mathematical problems arising from both the calculus of variations and control theory. A common feature of the problems is their affinity for the framework of viscosity solutions. The development of new techniques and methods necessary to resolve these problems will both advance the theory of viscosity solutions and provide new examples of their applicability. Viscosity solutions of differential equations were introduced in 1981 to characterize uniquely the value functions of optimal control and differential games. Part of the project will continue the study of such applications. Two other problem areas come from the calculus of variations. The first is based on the strategy of finding functions with prescribed boundary values on a domain which has minimal L-1 norm. These problems have been successfully studied using viscosity solutions. The current work will use viscosity solutions to analyze specific solutions of problems of motion by mean curvature to help understand the level sets of these functions. The second line of investigation is to study the same sets of functions as before with the goal of minimizing the maximal norm. This gives rise to fourth order homogeneous partial differential equations. Some work has already been done on the analysis of such questions but much more remains. 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. ***