Read more
This self-contained textbook provides the basic, abstract tools used in nonlinear analysis and their applications to semilinear elliptic boundary value problems. By first outlining the advantages and disadvantages of each method, this comprehensive text displays how various approaches can easily be applied to a range of model cases.
An Introduction to Nonlinear Functional Analysis and Elliptic Problems is divided into two parts: the first discusses key results such as the Banach contraction principle, a fixed point theorem for increasing operators, local and global inversion theory, Leray-Schauder degree, critical point theory, and bifurcation theory; the second part shows how these abstract results apply to Dirichlet elliptic boundary value problems. The exposition is driven by numerous prototype problems and exposes a variety of approaches to solving them.
Complete with a preliminary chapter, an appendix that includes further results on weak derivatives, and chapter-by-chapter exercises, this book is a practical text for an introductory course or seminar on nonlinear functional analysis.
List of contents
Notation.- Preliminaries.- Some Fixed Point Theorems.- Local and Global Inversion Theorems.- Leray-Schauder Topological Degree.- An Outline of Critical Points.- Bifurcation Theory.- Elliptic Problems and Functional Analysis.- Problems with A Priori Bounds.- Asymptotically Linear Problems.- Asymmetric Nonlinearities.- Superlinear Problems.- Quasilinear Problems.- Stationary States of Evolution Equations.- Appendix A Sobolev Spaces.- Exercises.- Index.- Bibliography.
About the author
Both authors are leading experts in this area of mathematics. Antonio Ambrosetti has been at the very forefront of research in this field for forty years, and several of the major topics from Parts 1 and 2 of the book are drawn from his research.
Summary
This self-contained textbook provides the basic, abstract tools used in nonlinear analysis and their applications to semilinear elliptic boundary value problems. By first outlining the advantages and disadvantages of each method, this comprehensive text displays how various approaches can easily be applied to a range of model cases.
An Introduction to Nonlinear Functional Analysis and Elliptic Problems is divided into two parts: the first discusses key results such as the Banach contraction principle, a fixed point theorem for increasing operators, local and global inversion theory, Leray–Schauder degree, critical point theory, and bifurcation theory; the second part shows how these abstract results apply to Dirichlet elliptic boundary value problems. The exposition is driven by numerous prototype problems and exposes a variety of approaches to solving them.
Complete with a preliminary chapter, an appendix that includes further results on weak derivatives, and chapter-by-chapter exercises, this book is a practical text for an introductory course or seminar on nonlinear functional analysis.
Additional text
From the reviews:
“The book is devoted to nonlinear functional analysis and its applications to semilinear elliptic boundary value problems. It covers a great variety of topics and gives a good introduction to the subject. … The book is aimed at graduate and senior undergraduate students.” (Alexander A. Pankov, Mathematical Reviews, Issue 2012 f)
“This book provides some basic abstract tools used in modern nonlinear analysis in strong relationship with their applications to semilinear elliptic boundary value problems. … This monograph is suitable for graduate students and researchers … . the volume under review should certainly be in the library of every university where research in mathematics is conducted.” (Vicenţiu D. Rădulescu, Zentralblatt MATH, Vol. 1228, 2012)
Report
From the reviews:
"The book is devoted to nonlinear functional analysis and its applications to semilinear elliptic boundary value problems. It covers a great variety of topics and gives a good introduction to the subject. ... The book is aimed at graduate and senior undergraduate students." (Alexander A. Pankov, Mathematical Reviews, Issue 2012 f)
"This book provides some basic abstract tools used in modern nonlinear analysis in strong relationship with their applications to semilinear elliptic boundary value problems. ... This monograph is suitable for graduate students and researchers ... . the volume under review should certainly be in the library of every university where research in mathematics is conducted." (Vicentiu D. Radulescu, Zentralblatt MATH, Vol. 1228, 2012)
