Nonlinear Functional Analysis and its Applications: III: Variational Methods and Optimization

✍ Scribed by Eberhard Zeidler (auth.)

• Publisher: Springer-Verlag New York
• Year: 1985
• Tongue: English
• Leaves: 675
• Edition: 1
• Category: Library

✦ Synopsis

As long as a branch of knowledge offers an abundance of problems, it is full of vitality. David Hilbert Over the last 15 years I have given lectures on a variety of problems in nonlinear functional analysis and its applications. In doing this, I have recommended to my students a number of excellent monographs devoted to specialized topics, but there was no complete survey-type exposition of nonlinear functional analysis making available a quick survey to the wide range of readers including mathematicians, natural scientists, and engineers who have only an elementary knowledge of linear functional analysis. I have tried to close this gap with my five-part lecture notes, the first three parts of which have been published in the Teubner-Texte series by Teubner-Verlag, Leipzig, 1976, 1977, and 1978. The present English edition was translated from a completely rewritten manuscript which is significantly longer than the original version in the Teubner-Texte series. The material is organized in the following way: Part I: Fixed Point Theorems. Part II: Monotone Operators. Part III: Variational Methods and Optimization. Parts IV jV: Applications to Mathematical Physics. The exposition is guided by the following considerations: (a) What are the supporting basic ideas and what intrinsic interrelations exist between them? (/3) In what relation do the basic ideas stand to the known propositions of classical analysis and linear functional analysis? ( y) What typical applications are there? Vll Preface viii Special emphasis is placed on motivation.

✦ Table of Contents

Front Matter....Pages i-xxii
Introduction to the Subject....Pages 1-11
Introductory Typical Examples....Pages 12-142
Front Matter....Pages 143-143
Compactness and Extremal Principles....Pages 145-167
Convexity and Extremal Principles....Pages 168-186
Front Matter....Pages 187-187
Free Local Extrema of Differentiable Functionals and the Calculus of Variations....Pages 189-228
Potential Operators....Pages 229-243
Free Minima for Convex Functionals, Ritz Method and the Gradient Method....Pages 244-269
Front Matter....Pages 271-272
Lagrange Multipliers and Eigenvalue Problems....Pages 273-312
Ljusternik-Schnirelman Theory and the Existence of Several Eigenvectors....Pages 313-350
Bifurcation for Potential Operators....Pages 351-360
Front Matter....Pages 361-361
Differentiable Functionals on Convex Sets....Pages 363-378
Convex Functionals on Convex Sets and Convex Analysis....Pages 379-406
General Lagrange Multipliers (Dubovickii-Miljutin Theory)....Pages 407-452
Front Matter....Pages 453-455
General Duality Principle by Means of Lagrange Functions and Their Saddle Points....Pages 457-478
Duality and the Generalized Kuhn-Tucker Theory....Pages 479-486
Duality, Conjugate Functionals, Monotone Operators and Elliptic Differential Equations....Pages 487-511
General Duality Principle by Means of Perturbed Problems and Conjugate Functionals....Pages 512-537
Conjugate Functionals and Orlicz Spaces....Pages 538-546
Front Matter....Pages 547-549
Elliptic Variational Inequalities....Pages 551-567
Evolution Variational Inequalities of First Order in H-Spaces....Pages 568-576
Front Matter....Pages 547-549
Evolution Variational Inequalities of Second Order in H-Spaces....Pages 577-580
Accretive Operators and Multivalued First-Order Evolution Equations in B-Spaces....Pages 581-598
Back Matter....Pages 599-662

✦ Subjects

Systems Theory, Control; Calculus of Variations and Optimal Control; Optimization; Analysis