Sinossi
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.
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Recensione
"... the book is a valuable contribution to the literature. It is well-written, self-contained and it has an extensive bibliography, especially with regard to the literature in the Russian language."
(Mathematical Reviews, 2001a)
Contenuti
Preface. 1. Preliminaries. 2. Minimization of Nonlinear Functionals. 3. Homotopic Methods in Variational Problems. 4. Topological Characteristics of Extremals of Variational Problems. 5. Applications. Bibliographical Comments. References. Index.
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Taschenbuch. Condizione: Neu. This item is printed on demand - it takes 3-4 days longer - Neuware -Since the building of all the Universe is perfect and is cre ated by the wisdom Creator, nothing arises in the Universe in which one cannot see the sense of some maXImum or mInImUm Euler God moves the Universe along geometrical lines Plato Mathematical models of most closed physical systems are based on vari ational principles, i.e., it is postulated that equations describing the evolu tion of a system are the Euler~Lagrange equations of a certain functional. In this connection, variational methods are one of the basic tools for studying many problems of natural sciences. The first problems related to the search for extrema appeared as far back as in ancient mathematics. They go back to Archimedes, Appolonius, and Euclid. In many respects, the problems of seeking maxima and minima have stimulated the creation of differential calculus; the variational prin ciples of optics and mechanics, which were discovered in the seventeenth and eighteenth centuries, gave impetus to an intensive development of the calculus of variations. In one way or another, variational problems were of interest to such giants of natural sciences as Fermat, Newton, Descartes, Euler, Huygens, 1. Bernoulli, J. Bernoulli, Legendre, Jacobi, Kepler, La grange, and Weierstrass. 564 pp. Englisch. Codice articolo 9789401059558
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Condizione: New. Dieser Artikel ist ein Print on Demand Artikel und wird nach Ihrer Bestellung fuer Sie gedruckt. Preface. 1. Preliminaries. 2. Minimization of Nonlinear Functionals. 3. Homotopic Methods in Variational Problems. 4. Topological Characteristics of Extremals of Variational Problems. 5. Applications. Bibliographical Comments. References. Index. . Codice articolo 5832591
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Taschenbuch. Condizione: Neu. Geometrical Methods in Variational Problems | N. A. Bobylov (u. a.) | Taschenbuch | xvi | Englisch | 2012 | Springer | EAN 9789401059558 | Verantwortliche Person für die EU: Springer Verlag GmbH, Tiergartenstr. 17, 69121 Heidelberg, juergen[dot]hartmann[at]springer[dot]com | Anbieter: preigu Print on Demand. Codice articolo 105565518
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Taschenbuch. Condizione: Neu. This item is printed on demand - Print on Demand Titel. Neuware -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.Springer-Verlag GmbH, Tiergartenstr. 17, 69121 Heidelberg 564 pp. Englisch. Codice articolo 9789401059558
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