โ Scribed by Fabio Silva Botelho
No coin nor oath required. For personal study only.
The book discusses basic concepts of functional analysis, measure and integration theory, calculus of variations and duality and its applications to variational problems of non-convex nature, such as the Ginzburg-Landau system in superconductivity, shape optimization models, dual variational formulations for micro-magnetism and others. Numerical Methods for such and similar problems, such as models in flight mechanics and the Navier-Stokes system in fluid mechanics have been developed through the generalized method of lines, including their matrix finite dimensional approximations. It concludes with a review of recent research on Riemannian geometry applied to Quantum Mechanics and Relativity. The book will be of interest to applied mathematicians and graduate students in applied mathematics. Physicists, engineers and researchers in related fields will also find the book useful in providing a mathematical background applicable to their respective professional areas.
Preface
Acknowledgements
Contents
SECTION I: FUNCTIONAL ANALYSIS
1. Metric Spaces
2. Topological Vector Spaces
3. Hilbert Spaces
4. The Hahn-Banach Theorems and the Weak Topologies
5. Topics on Linear Operators
6. Spectral Analysis, a General Approach in Normed Spaces
7. Basic Results on Measure and Integration
8. The Lebesgue Measure in Rn
9. Other Topics in Measure and Integration
10. Distributions
11. The Lebesgue and Sobolev Spaces
SECTION II: CALCULUS OF VARIATIONS, CONVEX ANALYSIS AND RESTRICTED OPTIMIZATION
12. Basic Topics on the Calculus of Variations
13. More Topics on the Calculus of Variations
14. Convex Analysis and Duality Theory
15. Constrained Variational Optimization
16. On Central Fields in the Calculus of Variations
SECTION III: APPLICATIONS TO MODELS IN PHYSICS AND ENGINEERING
17. Global Existence Results and Duality for Non-Linear Models of Plates and Shells
18. A Primal Dual Formulation and a Multi-Duality Principle for a Non-Linear Model of Plates
19. On Duality Principles for One and Three-Dimensional Non-Linear Models in Elasticity
20. A Primal Dual Variational Formulation Suitable for a Large Class of Non-Convex Problems in Optimization
21. A Duality Principle and Concerning Computational Method for a Class of Optimal Design Problems in Elasticity
22. Existence and Duality Principles for the Ginzburg-Landau System in Superconductivity
23. Existence of Solution for an Optimal Control Problem Associated to the Ginzburg-Landau System in Superconductivity
24. Duality for a Semi-Linear Model in Micro-Magnetism
25. About Numerical Methods for Ordinary and Partial Differential Equations
26. On the Numerical Solution of First Order Ordinary Differential Equation Systems
27. On the Generalized Method of Lines and its Proximal Explicit and Hyper-Finite Difference Approaches
28. On the Generalized Method of Lines Applied to the Time-Independent Incompressible Navier-Stokes System
29. A Numerical Method for an Inverse Optimization Problem through the Generalized Method of Lines
30. A Variational Formulation for Relativistic Mechanics based on Riemannian Geometry and its Application to the Quantum Mechanics Context
References
Index
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