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

9300966 Jensen这个项目将研究由变分演算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黏度解的框架具有亲和力。解决这些问题所必需的新技术和新方法的发展，将推动粘度解理论的发展，并提供其适用性的新实例。1981年引入了微分方程的粘度解，以独特地表征最优控制和微分对策的值函数。该项目的一部分将继续研究这些应用。另外两个问题来自于变分法。第一种方法是在具有最小L-1范数的定义域上寻找具有规定边值的函数。这些问题已经用粘度溶液成功地研究过了。目前的工作将使用粘度解来分析平均曲率运动问题的具体解，以帮助理解这些函数的水平集。研究的第二条线是研究与之前相同的函数集，目标是最小化最大范数。这就得到了四阶齐次偏微分方程。对这些问题的分析已经做了一些工作，但还需要做更多的工作。偏微分方程是物理世界数学建模的基础。数学分析的作用与其说是创建方程，不如说是提供关于解的定性和定量信息。这可能包括回答关于独特性、平滑性和增长性的问题。此外，分析常常发展出解的近似方法和对这些近似精度的估计。* * *

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. ***