题 目：Maximum Principles and Their Applications

报告人：Dr. Cristian Enache Ovidius University of Constanta, Romania

时 间：从10月15日开始，每周六上午9:00-11:30，共8次

地 点：管理科研楼 1418

对 象：大二以上学生

摘 要：In this course we will discuss about the maximum principles and their applications to the study of partial differential equations. More exactly, we will show how we employ the maximum principles to obtain information about uniqueness, approximation, boundedness, convexity, symmetry or asymtotic behaviour of solutions, without any explicit knowledge of solutions themselves.

The course will be organized in two parts.

The purpose of the part is to briefly introduce the terminology and the main tools that will be used throughout the course. We will start by introducing the second order linear differential operators of elliptic and parabolic type. Then, we will develop the first and second maximum principles of E. Hopf for elliptic equations and the maximum principles of L. Nirenberg and A. Friedman for parabolic equations. Moreover, we will present a short introduction to the normal coordinates, which are widely used in the literature, but curiously ignored in the mathematical textbooks.

The second part is concerned with the application of the maximum principles introduced in the first part of the course. We will introduce various P(ayne)-functions, which are nothing else than appropriate functional combinations of the solution and their derivatives, and derive new maximum principles for such functional. Moreover, We will show how to employee these new maximum principles to get a priori estimates, isoperimetric inequalities, symmetry and convexity results and Liouville type theorems in the elliptic case or spatial and temporal asymptotic behaviour of solutions in the parabolic case.

References:

M. H. PROTTER, H. F. WEINBERGER, Maximum principles in differential equations, Springer Verlag, (1975).

R. P. SPERB, Maximum Principles and Their Applications, Academic Press, (1981).

L.E. FRAENKEL, Introduction to Maximum Principles and Symmetry, Cambridge University Press, (2000).