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

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ABSTRACT The principal investigator proposes to study three problems inthe area of partial differential equations, namely the nonlineartwo-phase elliptic free boundary problem, symmetrization for diffusionequations with bounded measurable coefficients and symmetric behaviorof solutions of nonlinear parabolic equations. These problems defy the standard treatments which succeed in solving linear problems orequations in divergence form because of their nonlinear or non-divergentstructural nature. Therefore, solution of these problems will notonly 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 orequations 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 partialdifferential equations and diffusion processes becomes increasinglyimportant in the area of partial differential equations. The problemsstudied by the principal investigator in this project motivate ideasthat lead to the creation of some methods that might succeed in treating other partial differential equation problems as well. Solvingthese problems gives affirmative answer to the question of the well-posedness of these mathematical models which originate in physics andother disciplines. In addition, the theoretical treatment of theseproblems will shed light on the numerical solution of these and relatedproblems. Taking into account the origins of these problems, one isliable to believe applications of the theory about these problems canbe sought in disciplines other than mathematics.