The book has been mostly rewritten to bring in various improvements
and additions. In particular, the local theory is replaced with a global
treatment based on simple ideas of convexity and monotone operators.
Another major change is that the class of problems treated is much wider
than the Dirichlet type originally discussed. In addition, the variational
results are given a geometrical formulation that includes the hypercircle,
and error estimates for variational solutions are also described.
The number of applications to linear and nonlinear boundary value
problems has been doubled, covering some thirty cases which arise in
mathematical physics, chemistry, engineering, and biology. As well as
containing new derivations of well-known results such as the Rayleigh
and Temple bounds for eigenvalues, the examples contain many results
on upper and lower bounds that have only recently been obtained.
The book is written at a fairly elementary level and should be accessible
to any student with a little knowledge of the calculus of variations and
differential equations.
Author(s): A. M. Arthurs
Series: Oxford Mathematical Monographs
Edition: 2nd
Publisher: Clarendon Press
Year: 1980
Language: English
Pages: C+vii+154+B
Cover
OXFORD MATHEMATICAL MONOGRAPHS
COMPLEMENTARY VARIATIONAL PRINCIPLES
Copyright
Oxford University Press 1980
ISBN 0-19-853532-5
515'.62 QA379
LCCN 80-0613
PREFACE
CONTENTS
1 VARIATIONAL PRINCIPLES: INTRODUCTION
1.1. Introduction
1.2. Euler-Lagrange theory
1.3. Canonical formalism
1.4 Convex functions
1.5. Complementary variational principles
2 VARIATIONAL PRINCIPLES: SOME EXTENSIONS
2.1. A class of operators
2.2. Functional derivatives
2.3. Euler-Lagrange theory
2.4. Canonical formalism
2.5. Convex functionals
2.6. Complementary variational principles
3 LINEAR BOUNDARY-VALUE PROBLEMS
3.1. The inverse problem
3.2. A lass of linear problems
3.3. Variational formulation
3.4. Complementary principles
3.5. The hypercirde
3.6. Error estimates for approximate solutions
3.7. Alternative complementary principles
3.8. Estes for linear functionals
4 LINEAR APPLICATIONS
4.1. The Rayleigh and Temple bounds
4.2. Potential theory
4.3. Electrostatics
4.4. Diffusion
4.5. The Mime problem
4.6. Membrane with elastic support
4.7. Perturbation theory
4.8. Potential scattering
4.9. Other applications
5 NONLINEAR BOUNDARY-VALUE PROBLEMS
5.1. Class of problems
5.2. Variational formulation
5.3. Complementary principles
5.4. Monotone problems
5.5. Error estimates
5.6. Hypercircle results for monotone problems
5.7. Geometry of the general problem
5.8. Estimates for linear functionals
6 NONLINEAR APPLICATIONS
6.1. Poisson-Boltzmann equation
6.2. 17wmas.-Fermi equation
6.3. F8pp1-Henclcy equation
6.4. Prismatic bar
6.5. An integrral equation
6.6. Nonlinear diffusion
6.7. Nerve membrane problem
6.8. Nonlinear networks
6.9. Other applications
Conduding remarks
REFERENCES
SUBJECT INDEX
Back Cover
