An Introduction to Nonlinear Functional Analysis and Elliptic Problems

✍ Scribed by Antonio Ambrosetti, David Arcoya (auth.)

• Publisher: Birkhäuser Basel
• Year: 2011
• Tongue: English
• Leaves: 212
• Series: Progress in Nonlinear Differential Equations and Their Applications 82
• Edition: 1
• Category: Library

✦ Synopsis

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.

✦ Table of Contents

Front Matter....Pages 1-1
Preliminaries....Pages 1-15
Some Fixed Point Theorems....Pages 17-21
Local and Global Inversion Theorems....Pages 23-31
Leray—Schauder Topological Degree....Pages 33-45
An Outline of Critical Points....Pages 47-60
Bifurcation Theory....Pages 61-72
Elliptic Problems and Functional Analysis....Pages 73-82
Problems with A Priori Bounds....Pages 83-96
Asymptotically Linear Problems....Pages 97-110
Asymmetric Nonlinearities....Pages 111-119
Superlinear Problems....Pages 121-129
Quasilinear Problems....Pages 131-147
Stationary States of Evolution Equations....Pages 149-168
Back Matter....Pages 163-163

✦ Subjects

Functional Analysis; Partial Differential Equations; Dynamical Systems and Ergodic Theory