Mathematical Sciences: Second Order Elliptic and Parabolic Differential Equations

数学科学：二阶椭圆和抛物型微分方程

• 批准号: 9623287
• 负责人: Mikhail Safonov
• 金额: $ 11.17万
• 依托单位: University of Minnesota-Twin Cities
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 1996
• 资助国家: 美国
• 起止时间: 1996-06-15 至 1999-09-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=9623287&HistoricalAwards=false
• 关键词: Mathematical; Sciences; Second; Order; Elliptic

ABSTRACT DMS-9623287 PI: SAFONOV UNIVERSITY OF MINNESOTA The project relates to important topics in the theory of Second Order Elliptic and Parabolic Partial Differential Equations, such as properties of solutions of linear equations with measurable coefficients (the backward Harnack inequality, the estimates for fundamental solutions, etc.), the problem of uniqueness of blowup solutions to semilinear elliptic equations, and the problem of regularity of solutions to fully nonlinear equations. Part 1 of the project deals with the estimates of solutions of linear equations which do not depend on the smoothness of coefficients. Special attention is paid to the local boundedness of solutions and to the behavior of positive solutions which blowup at the boundary. Uniqueness of such solutions is investigated under minimal assumptions on the smoothness of coefficients and on the structure of the boundary. In Part 3, the results on the classical solutions of linear equations are extended to the fully nonlinear equations satisfying a Dini condition with respect to independent variables. The estimates of solutions of linear equations with non-smooth coefficients are associated with certain properties of different processes in non-homogeneous environment, with many applications to heat-mass transfer, chemistry, porous media, traffic flow, biology, etc. Such estimates also serve as a background for the theory of nonlinear equations. The investigation of blowup solutions of semilinear equations is important in ecology, combustion theory, chemical and nuclear engineering. Many phenomena in these and other areas can be treated in terms of a superdiffusion process which describes a cloud rising as the limit of a system of independent Brownian particles which die at random times, leaving a random number of offspring. Fully nonlinear equations are closely related to the theory of optimal control of random processes where the optimal strategy can e found by solving of appropriate initial or bou ndary value problem for the corresponding nonlinear equation. This theory has applications to many physical and economical problems.

摘要 DMS-9623287 PI：明尼苏达萨福诺夫大学 该项目涉及二阶椭圆和抛物型偏微分方程理论中的重要主题，例如具有可测量系数的线性方程解的性质（向后 Harnack 不等式、基本方程的估计）解等）、半线性椭圆方程的爆炸解的唯一性问题以及解的正则性问题完全非线性方程。 该项目的第 1 部分涉及不依赖于系数平滑度的线性方程解的估计。 特别关注解的局部有界性以及正解在边界处爆炸的行为。 在对系数平滑度和边界结构的最小假设下研究此类解的唯一性。 在第 3 部分中，线性方程经典解的结果被扩展到满足自变量 Dini 条件的完全非线性方程。 具有非光滑系数的线性方程解的估计与非均匀环境中不同过程的某些性质相关，在热质传递、化学、多孔介质、交通流、生物学等方面有许多应用。此类估计还作为非线性方程理论的背景。 半线性方程爆炸解的研究在生态学、燃烧理论、化学和核工程中具有重要意义。 这些和其他领域的许多现象都可以用超扩散过程来处理，该过程将云的上升描述为独立布朗粒子系统的极限，这些粒子在随机时间死亡，留下随机数量的后代。 全非线性方程与随机过程的最优控制理论密切相关，其中可以通过求解相应非线性方程的适当初始或边界值问题来找到最优策略。 该理论可应用于许多物理和经济问题。