Spectral Theory and Nonlinear Functional Analysis

ISBN-13: 9781584882497 / Angielski / Miękka / 2001 / 280 str.

Julian Lopez-Gomez

Opis

This Research Note introduces a new Morse (index) theory that is easie r to use, less technical, and more flexible than existing theories. Un til now, these results have appeared only in scattered journals. The a uthor provides a self-contained treatment with full proofs of the majo r results and complete with applications of the theory to Hamiltonian systems and other areas of current research, including wave equations, Chern-Simon functionals, and Lorentzian geometry

Recenzje

"The book is both an excellent introduction to some novel ideas about nonlinear eigenvalue problems and an exposition of a range of earlier results scattered in different papers and expounded in book form for the first time here."
- Mathematical Reviews, Issue 2002

"This is a nice introductory text about classical functional analysis…" This book will be interesting and useful for many mathematicians, scientists, graduate and undergraduate students."
-Mathematical Reviews

Spis treści

INTRODUCTION
General Assumptions and Basic Concepts
Some New Results
Historical Remarks
BIFURCATION FROM SIMPLE EIGENVALUES
Simple Eigenvalues and Transversality
The Theorem of M.G. Crandall and P.H. Rabinowitz
Local Bifurcation Diagrams
The Exchange Stability Principle
Applications
FIRST GENERAL BIFURCATION RESULTS
Lyapunov-Schmidt Reductions
The theorem of J. Ize
The Global Alternative of P.H. Rabinowitz
The Theorem of D. Westreich
THE ALGEBRAIC MULTIPLICITY
Motivating the Concept of Transversality
Transversal Eigenvalues
Algebraic Eigenvalues
Analytic Families
Simple Degenerate Eigenvalues
FUNDAMENTAL PROPERTIES OF THE MULTIPLICITY
The Multiplicity of R.J. Magnus
Relations between c and m
The Fundamental Theorem
The Classical Algebraic Multiplicity
Finite Dimensional Characterizations
The Parity of the Crossing Number
GLOBAL BIFURCATION THEORY
Preliminaries
Local Bifurcation
Global Behavior of the Bounded Components
Unilateral Global Bifurcation
Unilateral Bifurcation for Positive Operators
APPLICATIONS
Positive Solutions o Semilinear Elliptic Problems
Coexistence States for Elliptic Systems
Examples
A Further Application
REFERENCES
INDEX

O autorze

Julian Lopez-Gomez