.

0 (hodnocen0 x )

(0.8) Půjčeno:5x 

BK

1st ed.

Cambridge : Cambridge University, 1999

xvi,323 s.

ISBN 0-521-64203-5 (váz.)

Cambridge studies in advanced mathematics ; [vol.] 64

Obsahuje předmluvu, dodatky, rejstřík

Počet variační - pojednání

000011577

Paperback Re-issue // This textbook on the calculus of variations leads the reader from the basics to modem aspects of the theory. One-dimensional problems and classical issues like Euler-Lagrange equations are treated, as are Noether’s theorem, Hamilton-Jacobi theory, and in particular geodesic lines, thereby developing some important geometric and topological aspects. The basic ideas of optimal control theory are also given. The second part of the book deals with multiple integrals. Alter a review of Lebesgue integration, Banach and Hilbert space theory and Sobolev spaces (with complete and detailed proofs), there is a treatment of the direct methods and the fundamental lower semicontinuity theorems. Subsequent chapters introduce the basic concepts of the modem calculus of variations, namely relaxation, Gamma convergence, bifurcation theory and minimax methods based on the Palais-Smale condition. The prerequisites are only the basic results from calculus of one and several variables. After having studied this book, the reader will be well equipped to read research papers in the calculus of variations. // Cambridge Studies in Advanced Mathematics // EDITORS // D. J. H. Garling University of Cambridge W. Fulton University of Chicago T. tom Dieck University of Gottingen P. Walters Warwick University // Cambridge // UNIVERSITY PRESS www.cambridge.org // ISBN 0-521-05712-4 // Preface and summary Remarks on notation // page x // XV // 1 // 1.1 // 1.2 // 1.3 // 1.4 // 1.5 // 2

2.1 // 2.2 // 2.3 // 3 // 3.1 // 3.2 // 4 // 4.1 // 4.2 // 4.3 // 4.4 // 4.5 // 4.6 // Part one: One-dimensional variational problems // The classical theory // The Euler-Lagrange equations. Examples // The idea of the direct methods and some regularity results // The second variation. Jacobi fields // Free boundary conditions // Symmetries and the theorem of E. Noether // A geometric example: geodesic curves The length and energy of curves Fields of geodesic curves The existence of geodesics // Saddle point constructions // A finite dimensional example // The construction of Lyusternik—Schnirelman // The theory of Hamilton and Jacobi // The canonical equations // The Hamilton-Jacobi equation // Geodesics // Fields of extremals // Hilbert’s invariant integral and Jacobi’s theorem Canonical transformations // 3 // 3 // 10 // 18 // 24 // 26 // 32 // 32 // 43 // 51 // 62 // 62 // 67 // 79 // 79 // 81 // 87 // 89 // 92 // 95 // viii Contents // 5 Dynamic optimization 104 // 5.1 Discrete control problems 104 // 5.2 Continuous control problems 106 // 5.3 The Pontryagin maximum principle 109 // Part two: Multiple integrals in the calculus of variations 115 // 1 Lebesgue measure and integration theory 117 // 1.1 The Lebesgue measure and the Lebesgue integral 117 // 1.2 Convergence theorems 122 // 2 Banach spaces 125 // 2.1 Definition and basic properties of Banach and Hilbert // spaces 125 // 2.2 Dual spaces and weak convergence 132 // 2.3 Linear operators between Banach spaces 144

// 2.4 Calculus in Banach spaces 150 // 3 Lp and Sobolev spaces 159 // 3.1 Lp spaces 159 // 3.2 Approximation of Lp functions by smooth functions // (mollification) 166 // 3.3 Sobolev spaces 171 // 3.4 Rellich’s theorem and the Poincaré and Sobolev // inequalities 175 // 4 The direct methods in the calculus of variations 183 // 4.1 Description of the problem and its solution 183 // 4.2 Lower semicontinuity 184 // 4.3 The existence of minimizers for convex variational // problems 187 // 4.4 Convex functionals on Hilbert spaces and Moreau- // Yosida approximation 190 // 4.5 The Euler-Lagrange equations and regularity questions 195 // 5 Nonconvex functionals. Relaxation 205 // 5.1 Nonlower semicontinuous functionals and relaxation 205 // 5.2 Representation of relaxed functionals via convex // envelopes 213 // 6 F-convergence 225 // 6.1 The definition of F-convergence 225 // Contents ix // 6.2 Homogenization 231 // 6.3 Thin insulating layers 235 // 7 BV-functionals and ?-convergence: the example of // Modica and Mortola 241 // 7.1 The space ?V(fi) 241 // 7.2 The example of Modica-Mortola 248 // Appendix A The coarea formula 257 // Appendix ? The distance function from smooth hypersurfaces 262 // 8 Bifurcation theory 266 // 8.1 Bifurcation problems in the calculus of variations 266 // 8.2 The functional analytic approach to bifurcation theory 270 // 8.3 The existence of catenoids as an example of a bifurcation process 282 // 9 The Palais-Smale condition and unstable critical

points of variational problems 291 // 9.1 The Palais-Smale condition 291 // 9.2 The mountain pass theorem 301 // 9.3 Topological indices and critical points 306 // Index 319