Mathematical Sciences: Second Order Elliptic and Parabolic Differential Equations

摘要dms-9623287 pi：项目涉及二阶椭圆型和抛物型偏微分方程理论中的重要课题，如系数可测线性方程解的性质（倒哈纳克不等式、基本解的估计等）、半线性椭圆型方程爆破解的唯一性问题、全非线性方程解的正则性问题等。项目的第一部分涉及不依赖于系数平滑的线性方程的解的估计。特别注意解的局部有界性和在边界处爆破的正解的行为。在系数光滑性和边界结构的最小假设下，研究了这类解的唯一性。在第三部分，将经典线性方程解的结果推广到关于自变量的满足Dini条件的完全非线性方程。非光滑系数线性方程解的估计与非均匀环境中不同过程的某些性质有关，在传热传质、化学、多孔介质、交通流、生物学等领域有广泛的应用。这样的估计也可以作为非线性方程理论的背景。半线性方程爆破解的研究在生态学、燃烧理论、化学和核工程中具有重要意义。这些领域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。完全非线性方程与随机过程的最优控制理论密切相关，通过求解相应非线性方程的适当初始或边值问题可以找到最优策略。这一理论适用于许多物理和经济问题。

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.