Geometrical Methods in Variational Problems [electronic resource] /

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 -- 5.8 Bifurcation of Extremals of Variational Problems -- 5.9 Eigenvalues of Potential Operators -- 5.10 Additional Remarks -- Bibliographical Comments -- References.

Bibliographic Details

Main Authors:	Bobylev, N. A. author., Emel’yanov, S. V. author., Korovin, S. K. author., SpringerLink (Online service)
Format:	Texto biblioteca
Language:	eng
Published:	Dordrecht : Springer Netherlands : Imprint: Springer, 1999
Subjects:	Mathematics., Global analysis (Mathematics)., Manifolds (Mathematics)., Differential equations., Partial differential equations., Mathematical optimization., Calculus of variations., Calculus of Variations and Optimal Control; Optimization., Optimization., Global Analysis and Analysis on Manifolds., Partial Differential Equations., Ordinary Differential Equations.,
Online Access:	http://dx.doi.org/10.1007/978-94-011-4629-6
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