Fully Nonlinear Free Boundary Problems, Stochastic Symmetrization, and Asymptotic Symmetry of Parabolic Equations

University Of Chicago, Chicago IL

Investigators

Abstract

ABSTRACT The principal investigator proposes to study three problems in the area of partial differential equations, namely the nonlinear two-phase elliptic free boundary problem, symmetrization for diffusion equations with bounded measurable coefficients and symmetric behavior of solutions of nonlinear parabolic equations. These problems defy the standard treatments which succeed in solving linear problems or equations in divergence form because of their nonlinear or non-divergent structural nature. Therefore, solution of these problems will not only contributes knowledge about these partial differential equations, but also inspires new ideas which will most probably prove useful in studying other nonlinear partial differential equations or equations of other types. The suggested methods in this proposal either successfully solved part of the problems or are motivated by the ideas that are most likely to lead to the solution of these problems. The study of the theory of nonlinear elliptic and parabolic partial differential equations and diffusion processes becomes increasingly important in the area of partial differential equations. The problems studied by the principal investigator in this project motivate ideas that lead to the creation of some methods that might succeed in treating other partial differential equation problems as well. Solving these problems gives affirmative answer to the question of the well- posedness of these mathematical models which originate in physics and other disciplines. In addition, the theoretical treatment of these problems will shed light on the numerical solution of these and related problems. Taking into account the origins of these problems, one is liable to believe applications of the theory about these problems can be sought in disciplines other than mathematics.