Contents

Preface page xiii

1 Extrema 1

1.1 Introduction 1

1.2 A one dimensional problem 1

1.3 The Hilbert space H 10

1.4 Fourier series 17

1.5 Finding a functional 20

1.6 Finding a minimum, I 23

1.7 Finding a minimum, II 28

1.8 A slight improvement 30

1.9 Finding a minimum, III 32

1.10 The linear problem 33

1.11 Nontrivial solutions 35

1.12 Approximate extrema 36

1.13 The Palais-Smale condition 40

1.14 Exercises 42

2 Critical points 45

2.1 A simple problem 45

2.2 A critical point 46

2.3 Finding a Palais-Smale sequence 47

2.4 Pseudo-gradients 52

2.5 A sandwich theorem 55

2.6 A saddle point 60

2.7 The chain rule 64

vii

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0521843979 - An Introduction to Nonlinear Analysis

Martin Schechter

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2.8 The Banach fixed point theorem 65

2.9 Picard's theorem 66

2.10 Continuous dependence of solutions 68

2.11 Continuation of solutions 69

2.12 Extending solutions 71

2.13 Resonance 72

2.14 The question of nontriviality 75

2.15 The mountain pass method 76

2.16 Other intervals for asymptotic limits 79

2.17 Super-linear problems 82

2.18 A general mountain pass theorem 83

2.19 The Palais-Smale condition 85

2.20 Exercises 85

3 Boundary value problems 87

3.1 Introduction 87

3.2 The Dirichlet problem 87

3.3 Mollifiers 88

3.4 Test functions 90

3.5 Differentiability 92

3.6 The functional 99

3.7 Finding a minimum 101

3.8 Finding saddle points 107

3.9 Other intervals 110

3.10 Super-linear problems 114

3.11 More mountains 116

3.12 Satisfying the Palais-Smale condition 119

3.13 The linear problem 120

3.14 Exercises 121

4 Saddle points 123

4.1 Game theory 123

4.2 Saddle points 123

4.3 Convexity and lower semi-continuity 125

4.4 Existence of saddle points 128

4.5 Criteria for convexity 132

4.6 Partial derivatives 133

4.7 Nonexpansive operators 137

4.8 The implicit function theorem 139

4.9 Exercises 143

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0521843979 - An Introduction to Nonlinear Analysis

Martin Schechter

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Contents ix

5 Calculus of variations 145

5.1 Introduction 145

5.2 The force of gravity 145

5.3 Hamilton's principle 148

5.4 The Euler equations 151

5.5 The G^ateaux derivative 155

5.6 Independent variables 156

5.7 A useful lemma 158

5.8 Sufficient conditions 159

5.9 Examples 165

5.10 Exercises 167

6 Degree theory 171

6.1 The Brouwer degree 171

6.2 The Hilbert cube 175

6.3 The sandwich theorem 183

6.4 Sard's theorem 184

6.5 The degree for differentiable functions 187

6.6 The degree for continuous functions 193

6.7 The Leray-Schauder degree 197

6.8 Properties of the Leray-Schauder degree 200

6.9 Peano's theorem 201

6.10 An application 203

6.11 Exercises 205

7 Conditional extrema 207

7.1 Constraints 207

7.2 Lagrange multipliers 213

7.3 Bang-bang control 215

7.4 Rocket in orbit 217

7.5 A generalized derivative 220

7.6 The definition 221

7.7 The theorem 222

7.8 The proof 226

7.9 Finite subsidiary conditions 229

7.10 Exercises 235

8 Mini-max methods 237

8.1 Mini-max 237

8.2 An application 240

8.3 Exercises 243

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0521843979 - An Introduction to Nonlinear Analysis

Martin Schechter

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9 Jumping nonlinearities 245

9.1 The Dancer-Fu?cÃ¯Ã•k spectrum 245

9.2 An application 248

9.3 Exercises 251

10 Higher dimensions 253

10.1 Orientation 253

10.2 Periodic functions 253

10.3 The Hilbert spaces Ht 254

10.4 Compact embeddings 258

10.5 Inequalities 258

10.6 Linear problems 262

10.7 Nonlinear problems 265

10.8 Obtaining a minimum 271

10.9 Another condition 274

10.10 Nontrivial solutions 277

10.11 Another disappointment 278

10.12 The next eigenvalue 278

10.13 A Lipschitz condition 282

10.14 Splitting subspaces 283

10.15 The question of nontriviality 285

10.16 The mountains revisited 287

10.17 Other intervals between eigenvalues 289

10.18 An example 293

10.19 Satisfying the PS condition 294

10.20 More super-linear problems 297

10.21 Sobolev's inequalities 297

10.22 The case q = 8 303

10.23 Sobolev spaces 305

10.24 Exercises 308

Appendix A Concepts from functional analysis 313

A.1 Some basic definitions 313

A.2 Subspaces 314

A.3 Hilbert spaces 314

A.4 Bounded linear functionals 316

A.5 The dual space 317

A.6 Operators 319

A.7 Adjoints 321

A.8 Closed operators 322

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0521843979 - An Introduction to Nonlinear Analysis

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A.9 Self-adjoint operators 323

A.10 Subsets 325

A.11 Finite dimensional subspaces 326

A.12 Weak convergence 327

A.13 Reflexive spaces 328

A.14 Operators with closed ranges 329

Appendix B Measure and integration 331

B.1 Measure zero 331

B.2 Step functions 331

B.3 Integrable functions 332

B.4 Measurable functions 335

B.5 The spaces Lp 335

B.6 Measurable sets 336

B.7 CarathÃ¯eodory functions 338

Appendix C Metric spaces 341

C.1 Properties 341

C.2 Para-compact spaces 343

Appendix DPseudo-gradien ts 345

D.1 The benefits 345

D.2 The construction 346

Bibliography 353

Index 355