Crandall 9302995 This project continues mathematical research on nonlinear partial differential equations. A very broad class of equations referred to as fully nonlinear will be under consideration. Also included will be the associated evolution equation one can form from a time-dependent differential equation. The thrust of the work carries forward investigations into a class of solutions developed during the past decade known as viscosity solutions. To be precise, a viscosity solution is a class of functions within which one expects to find a solution to the equation. The methods are closely related to that of continuous game theory in that from many possible solutions one wants to show that a best can be determined. There has been considerable success in obtaining existence results for viscosity solutions in which the basic operators involved are bounded. The present work seeks to expand the possibilities for extending to unbounded versions and infinite dimensional generalizations. A second line of investigation will be carried out in an effort to obtain numerical approximations of fully nonlinear equations. The work proceeds in several directions. One must determine what an effective type mesh would be and in terms of this give error estimates. This will first be done without regard to boundary conditions, but with the understanding that such cases must be taken into account at a later date. A modest beginning will be made on uniformly elliptic equations. 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 numerical approximation of solutions and estimates on the accuracy of these approximations. ***
