Yu D. Order Structure and Topological Methods in Nonlinear PDE. Vol.1

World Scientific Publishing Co. Pte. Ltd., 2006. — 202 p. — ISBN: 981-256-624-4This is volume one of a two volume series. The intention is to provide a reference book for researchers in nonlinear partial differential equations and nonlinear functional analysis, especially for postgraduate students who want to be led to some of the current research topics. It could be used as a textbook for postgraduate students, either in formal classes or in working seminars.
In these two volumes, we attempt to use order structure as a thread to introduce the various versions of the maximum principles, the fixed point index theory, and the relevant part of critical point theory and Conley index theory. The emphasize is on their applications, and we try to demonstrate the usefulness of these tools by choosing applications to problems in partial differential equations that are of considerable concern of current research.
An important work in this direction is H. Amann's classical review article (SIAM Rev. 18 (1976), 620-709), which discussed the combination of order structure and fixed point index theory and its applications to various problems of nonlinear partial differential equations. Much progress has been made since this article. The fixed point index theory has been further developed and found important new applications in partial differential equations. Moreover, the order structure has since been successfully combined with critical point theory and Conley index theory to study various nonlinear partial differential equation problems. Furthermore, the classical maximum principle in partial differential equations has found new applications in several important problems. All these are scattered in research articles published in various professional journals, and most of them are still active topics of current research.
It is our hope that through these two volumes, we can present the reader in a somewhat systematic way some of the new progresses in these topics. As the title suggests, volume 1 mainly considers the maximum principles and their various applications in some of the current research topics. The topological methods will be discussed in volume 2.