Mathematical Sciences: Second Order Elliptic and Parabolic Differential Equations

摘要 DMS-9623287 PI：明尼苏达州萨福诺夫大学 该项目涉及二阶椭圆和抛物偏微分方程理论中的重要课题，例如具有可测系数的线性方程的解的性质（向后Harnack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和完全非线性方程解的正则性问题。 该项目的第1部分涉及不依赖于系数光滑性的线性方程的解的估计。 特别注意解的局部有界性和正解在边界处爆破的行为。 这种解决方案的uniformity的最小假设下的光滑系数和边界的结构进行了研究。 第三部分将线性方程经典解的结果推广到关于自变量满足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，在热质传递、化学、多孔介质、交通流、生物学等方面有着广泛的应用。 半线性方程爆破解的研究在生态学、燃烧理论、化学与核工程等领域具有重要意义。 在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，该过程描述了一个云的上升，作为一个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有着密切的联系，其中最优控制策略可以通过求解相应的非线性方程的适当初值或边界值问题而得到。 这一理论可应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.