Geometrical Methods in Variational Problems
| Main Authors: | , , |
|---|---|
| Format: | eBook |
| Language: | English |
| Published: |
Dordrecht
Springer Netherlands
1999, 1999
|
| Edition: | 1st ed. 1999 |
| Series: | Mathematics and Its Applications
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| Subjects: | |
| Online Access: | |
| Collection: | Springer Book Archives -2004 - Collection details see MPG.ReNa |
Table of Contents:
- 5.8 Bifurcation of Extremals of Variational Problems
- 5.9 Eigenvalues of Potential Operators
- 5.10 Additional Remarks
- Bibliographical Comments
- References
- 1 Preliminaries
- 1.1 Metric and Normed Spaces
- 1.2 Compactness
- 1.3 Linear Functional and Dual Spaces
- 1.4 Linear Operators
- 1.5 Nonlinear Operators and Functionals
- 1.6 Contraction Mapping Principle, Implicit Function Theorem, and Differential Equations on a Banach Space
- 2 Minimization of Nonlinear Functionals
- 2.1 Extrema of Smooth Functionals
- 2.2 Extremum of Lipschitzian and Convex Functionals
- 2.3 Weierstass Theorems
- 2.4 Monotonicity
- 2.5 Variational Principles
- 2.6 Additional Remarks
- 3 Homotopic Methods in Variational Problems
- 3.1 Deformations of Functionals on Hilbert Spaces
- 3.2 Deformations of Functionals on Banach Spaces
- 3.3 Global Deformations of Functionals
- 3.4 Deformation of Problems of the Calculus of Variations
- 3.5 Deformations of Lipschitzian Functions
- 3.6 Global Deformations of Lipschitzian Functions
- 3.7 Deformations of Mathematical Programming Problems
- 3.8 Deformations of Lipschitzian Functionals
- 3.9 Additional Remarks
- 4 Topological Characteristics of Extremals of Variational Problems
- 4.1 Smooth Manifolds and Differential Forms
- 4.2 Degree of Mapping
- 4.3 Rotation of Vector Fields in Finite-Dimensional Spaces
- 4.4 Vector Fields in Infinite-Dimensional Spaces
- 4.5 Computation of the Topological Index
- 4.6 Topological Index of Zero of an Isolated Minimum
- 4.7 Euler Characteristic and the Topological Index of an Isolated Critical Set
- 4.8 Topological Index of Extremals of Problems of the Calculus of Variations
- 4.9 Topological Index of Optimal Controls
- 4.10 Topological Characteristic s of Critical Points of Nonsmooth Functionals
- 4.11 Additional Remarks
- 5 Applications
- 5.1 Existence Theorems
- 5.2 Bounds of the Number of Solutions to Variational Problems
- 5.3 Applications of the Homotopic Method
- 5.4 Study of Degenerate Extremals
- 5.5 Morse Lemmas
- 5.6 Well-Posedness of Variational Problems. Ulam Problem
- 5.7 Gradient Procedures