Modeling and Convexity

Introduction ix

PART 1 MOTIVATION: EXAMPLES AND APPLICATIONS 1

Chapter 1 Curvilinear Continuous Media 3

1.1 One-dimensional curvilinear media 4

1.2 Supple membranes 22

Chapter 2 Unilateral System Dynamics 33

2.1 Dynamics of ideally flexible strings 34

2.2 Contact dynamics 40

Chapter 3 A Simplified Model of Fusion/Solidification 53

3.1 A simplified model of phase transition 53

Chapter 4 Minimization of a Non-Convex Function 61

4.1 Probabilities, convexity and global optimization 61

Chapter 5 Simple Models of Plasticity 69

5.1 Ideal elastoplasticity 72

PART 2 THEORETICAL ELEMENTS 77

Chapter 6 Elements of Set Theory 79

6.1 Elementary notions and operations on sets 80

6.2 The axiomof choice 83

6.3 Zorn's lemma 89

Chapter 7 Real Hilbert Spaces 97

7.1 Scalar product and norm 99

7.2 Bases anddimensions 107

7.3 Open sets and closed sets 114

7.4 Sequences 123

7.5 Linear functionals 137

7.6 Complete space 146

7.7 Orthogonal projection onto a vector subspace 160

7.8 Riesz's representationtheory 167

7.9 Weak topology 173

7.10 Separable spaces: Hilbert bases and series 184

Chapter 8 Convex Sets 201

8.1 Hyperplanes 201

8.2 Convexsets 208

8.3 Convexhulls 212

8.4 Orthogonal projection on a convex set 217

8.5 Separationtheorems 228

8.6 Convexcone 241

Chapter 9 Functionals on a Hilbert Space 253

9.1 Basic notions 254

9.2 Convexfunctionals 261

9.3 Semi-continuous functionals 271

9.4 Affine functionals 298

9.5 Convexification and LSC regularization 303

9.6 Conjugate functionals 320

9.7 Subdifferentiability 331

Chapter 10 Optimization 361

10.1 The optimization problem 361

10.2 Basic notions 362

10.3 Fundamental results 374

Chapter 11 Variational Problems 421

11.1 Fundamental notions 421

11.2 Zeros of operators 455

11.3 Variational inequations 463

11.4 Evolutionequations 469

Bibliography 487

Index 495