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.
