Table of Contents
Preface v
1 The theory and problems of calculus of variations 1
1.1 Introduction 1
1.2 Functionals 3
1.3 Typical examples 4
1.4 Mote examples 8
2 The Euler-Lagrange equation 13
2.1 The necessary condition for the extremal values of functions - a review 13
2.2 The derivation of the Euler-Lagrange equation 14
2.3 Boundary conditions 20
2.4 Examples of solving the Euler-Lagrange equations 21
Exercises 26
3 The necessary condition and the sufficient condition on extremal values of functionals 29
3.1 The extremal values of functions - a revisit 29
3.2 Second order variations 30
3.3 The Legendre-Hadamard condition 32
3.4 The Jacobi field 34
3.5 Conjugate points 37
Exercises 41
4 Strong minima and extremal fields 43
4.1 Strong minima and weak minima 43
4.2 A necessary condition for strong minimal value and the Weierstrass excess function 44
4.3 Extremal fields and strong minima 46
4.4 Mayer field, Hilbert's invariant integral 52
4.5 A sufficient condition for strong minima 54
4.6* The proof of Theorem 4.4 (for the case N > 1) 56
Exercises 59
5 The Hamilton-Jacobi theory 61
5.1 Eikonal and the Carathéodory system of equations 61
5.2 The Legendre transformation 62
5.3 The Hamilton system of equations 64
5.4 The Hamilton-Jacobi equation 67
5.5* Jacobi's Theorem 69
Exercises 72
6 Variational problems involving multivariate integrals 73
6.1 Derivation of the Euler-Lagrange equation 73
6.2 Boundary conditions 80
6.3 Second order variations 81
6.4 Jacobi fields 84
Exercises 87
7 Constrained variational problems 89
7.1 The isoperimetric problem 89
7.2 Pointwise constraints 94
7.3 Variational inequalities 100
Exercises 101
8 The conservation law and Noether's theorem 103
8.1 One parameter diffeomorphisms and Noether's theorem 103
8.2 The energy-momentum tensor and Noether's theorem 107
8.3 Interior minima 112
8.4* Applications 115
Exercises 117
9 Direct methods 119
9.1 The Dirichlet's principle and minimization method 119
9.2 Weak convergence and weak-* convergence 122
9.3 Weak-* sequential compactness 125
9.4* Reflexive spaces and the Eberlein-Šmulian theorem 129
Exercises 132
10 Sobolev spaces 133
10.1 Generalized derivatives 133
10.2 The space Wm,p(ω) 134
10.3 Representations of functionals 137
10.4 Modifiers 138
10.5 Some important properties of Sobolev spaces and embedding theorems 139
10.6 The Euler-Lagrange equation 145
Exercises 148
11 Weak lower semi-continuity 149
11.1 Convex sets and convex functions 149
11.2 Convexity and weak lower semi-continuity 151
11.3 An existence theorem 154
11.4* Quasi-convexity 155
Exercises 161
12 Boundary value problems and eigenvalue problems of linear differential equations 163
12.1 Linear boundary value problems and orthogonal projections… 163
12.2 The eigenvalue problems 167
12.3 The eigenfunction expansions 171
12.4 The minimax description of eigenvalues 176
Exercises 177
13 Existence and regularity 179
13.1 Regularity (n = 1) 180
13.2 More on regularity (n > 1) 184
13.3 The solutions of some variational problems 186
13.4 The limitations of calculus of variations 193
Exercises 194
14 The dual least action principle and the Ekeland variational principle 195
14.1 The conjugate function of a convex function 195
14.2 The dual least action principle 199
14.3 The Ekeland variational principle 202
14.4 The Fréchet derivative and the Palais-Smale condition 203
14.5 The Nehari technique 206
Exercises 208
15 The Mountain Pass Theorem, its generalizations, and applications 211
15.1 The Mountain Pass Theorem 211
15.2 Applications 219
16 Periodic solutions, homoclinic and heteroclinic orbits 227
16.1 The simple pendulum 227
16.2 Periodic solutions 230
16.3 Heteroclinic orbits 234
16.4 Homoclinic orbits 238
17 Geodesies and minimal surfaces 243
17.1 Geodesies 243
17.2 Minimal surfaces 247
18 Numerical methods for variational problems 259
18.1 The Ritz method 259
18.2 The finite element method 261
18.3 Cea's theorem 266
18.4 An optimization method - the conjugate gradient method … 268
19 Optimal control problems 275
19.1 The formulation of problems 275
19.2 The Pontryagin Maximal Principle 280
19.3 The Bang-Bang principle 285
20 Functions of bounded variations and image processing 289
20.1 Functions of bounded variations in one variable -a review… 289
20.2 Functions of bounded variations in several variables 293
20.3 The relaxation function 299
20.4 Image restoration and the Rudin-Osher-Fatemi model 301
Bibliography 305
Index 309
