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

主要研究偏微分方程领域的三个问题，即非线性两相椭圆自由边界问题、系数有界扩散方程的对称化和非线性抛物型方程解的对称性。这些问题由于其非线性或非发散的结构性质，与成功地解决发散形式的线性问题的标准处理方法背道而驰。因此，这些问题的解不仅能提供关于这些偏微分方程解的知识，而且能启发新的思想，这些新思想很可能对研究其他类型的非线性偏微分方程解是有用的。本提案中建议的方法要么成功地解决了部分问题，要么受到最有可能导致这些问题解决的想法的推动。非线性椭圆型和抛物型偏微分方程及扩散过程理论的研究在偏微分方程界变得越来越重要。这个项目的主要研究人员所研究的问题激发了一些想法，这些想法导致了一些方法的创造，这些方法也可能成功地处理其他偏微分方程问题。解决这些问题，肯定地回答了起源于物理学和其他学科的这些数学模型的适定性问题。此外，对这些问题的理论处理将有助于这些问题和相关问题的数值解。考虑到这些问题的起源，人们很容易相信关于这些问题的理论可以在数学以外的学科中寻求应用。

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.