Geometrical Methods in Variational Problems

This self-contained monograph presents methods for the investigation of nonlinear variational problems. These methods are based on geometric and topological ideas such as topological index, degree of a mapping, Morse-Conley index, Euler characteristics, deformation invariant, homotopic invariant, and the Lusternik-Shnirelman category. Attention is also given to applications in optimisation, mathematical physics, control, and numerical methods.
Audience: This volume will be of interest to specialists in functional analysis and its applications, and can also be recommended as a text for graduate and postgraduate-level courses in these fields.

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.