Mathematical Sciences: Viscosity Solutions of Nonlinear Partial Differential Equations

本工作的重点是研究一类完全非线性偏微分方程解的性质。一种关于二阶方程这些解(称为粘性解)的综合理论正在迅速发展。取得了唯一性、规律性和可比性。在这个项目上要做的工作将继续朝着获得完全非线性方程粘性解的最大值原理的推广的方向发展。粘性解是弱解的一种，1983年首次引入一阶方程的粘性解。这个概念是由弱极大值原理产生的，这一原理使它有别于其他弱解的定义，如依赖于部分积分的定义。当粘性解存在时，它们总是与经典解一致。另一方面，它们可能只是半连续的，不可微的。当考虑解序列的极限时，这种性质特别有用。因为这些解是通过粘度为零的方法得到的，所以在适当的情况下，它们也声称是正确的物理解。本工作将继续研究应用于粘性解的经典最大值原理的不同版本和扩展，包括涉及不连续边界数据的问题。还将努力处理在几种情况下自然出现的奇异扰动，例如由具有风险厌恶效用函数的随机优化问题确定的微分对策。其他的应用也将应用于随机控制中的问题。两个具体领域包括描述在扩散过程和期权定价和风险厌恶金融系统中产生的问题之间跳跃的随机过程的问题。

The focus of this work will be on the investigation of a class of solutions of fully nonlinear partial differential equations. A comprehensive theory of these solutions (known as viscosity solutions) for second order equations is rapidly evolving. Uniqueness, regularity and comparison results have been achieved. Work to be done on this project will continue the development of the theory in the direction of obtaining extensions of the maximum principle for viscosity solutions of the fully nonlinear equations. Viscosity solutions, a type of weak solution, were first introduced in 1983 for first order equations. The concept is motivated by the weak maximum principle which distinguishes it from other definitions of weak solution such as those depending on integration by parts. Viscosity solutions always agree with classical solutions when they exist. On the other hand, it is possible that they may only be semicontinuous and not differentiable. Such properties are particularly useful when one is considering limits of sequences of solutions. Because these solutions are obtained by a method of vanishing viscosity, they also lay claim to being the correct physical solution where appropriate. The present work will continue studies of different versions and extensions of the classical maximum principle applied to viscosity solutions, including problems involving discontinuous boundary data. Efforts will also be made in treating singular perturbations which arise naturally in several contexts such as differential games determined by stochastic optimization problems with risk averse utility functions. Other applications will be made to problems in stochastic control. Two specific areas include the problem of characterizing a stochastic process which jumps between diffusion processes and questions arising in option pricing and risk averse financial systems.