9706497 Gutierrez The problems in this project concentrate on the study of the behavior of solutions of degenerate elliptic and parabolic equations in non-divergence form with non-smooth coefficients. Such coefficients may either vanish, be infinite, or both. A primary goal is to study the validity of the Harnack principle for nonnegative solutions and estimates of the second derivatives of solutions. Though the theory for divergence form equations with measurable coefficients has been the focus of research during recent years, the corresponding theory for non-divergence form equations is much less developed. Non-divergence equations are natural in Probability, Control Theory, and Finance. The results known in this direction concern either smooth coefficients or the strictly elliptic case, but the methods developed for those cases do not apply to the problems proposed. In some cases the geometry needed to study the linear equation is given by the Monge-Ampere equation, a fully nonlinear partial differential equation. This approach requires the study of the shape and invariance properties of the level sets of solutions to the Monge-Ampere equation, and further work will continue along these lines. The basis and methodology that will be used to solve this set of problems are via the maximum principle, localization, and nonlinear variants of the Calderon-Zygmund decomposition. This mathematical research is in the field of partial differential equations, linear and nonlinear. These equations are the principal classical tool of the applications of mathematics to the physical world. The project has a strong connection with Harmonic Analysis in Euclidean space, a subject that has flourished during the second half of the twentieth century and that has become an indispensable tool to provide qualitative and quantitative information about the solutions of partial differential equations. Some equations considered in the project are used in models of atmospheric a nd oceanic flows and in the description of the diffusion of a gas in a porous medium.
