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

完全非线性自由边界问题、抛物方程的随机对称性和渐近对称性

• 批准号: 0088973
• 负责人: Peiyong Wang
• 金额: $ 6.15万
• 依托单位: University of Chicago
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2000
• 资助国家: 美国
• 起止时间: 2000-07-15 至 2001-10-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0088973&HistoricalAwards=false
• 关键词: Fully; Nonlinear; Free; Boundary; Problems

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.

摘要 本文主要研究偏微分方程领域中的三个问题，即非线性两相椭圆自由边界问题、具有有界可测系数的扩散方程的对称化和非线性抛物方程解的对称性。这些问题无视标准的治疗方法，成功地解决线性问题或方程的发散形式，因为他们的非线性或non-divergentstructure性质。因此，这些问题的解决不仅有助于了解这些偏微分方程，而且还启发了新的想法，这将是最有可能证明有用的研究其他非线性偏微分方程或其他类型的方程。本提案中建议的方法要么成功地解决了部分问题，要么是受到最有可能导致解决这些问题的想法的激励。 非线性椭圆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领域中越来越重要。在这个项目中，主要研究者研究的问题激发了一些想法，这些想法导致了一些方法的产生，这些方法也可能成功地处理其他偏微分方程问题。这些问题的解决，肯定地回答了这些数学模型的适定性问题，这些数学模型起源于物理学和其他学科。此外，这些问题的理论处理将阐明这些问题和相关问题的数值解。考虑到这些问题的起源，人们倾向于相信，可以在数学以外的学科中寻求关于这些问题的理论的应用。