Co-Semigroups and Applications (eBook)

Cover 1
Contents 7
Preface 11
Chapter 1. Preliminaries 13
1.1. Vector-Valued Measurable Functions 13
1.2. The Bochner Integral 16
1.3. Basic Function Spaces 21
1.4. Functions of Bounded Variation 24
1.5. Sobolev Spaces 27
1.6. Unbounded Linear Operators 32
1.7. Elements of Spectral Analysis 36
1.8. Functional Calculus for Bounded Operators 39
1.9. Functional Calculus for Unbounded Operators 43
Problems 45
Notes 46
Chapter 2. Semigroups of Linear Operators 47
2.1. Uniformly Continuous Semigroups 47
2.2. Generators of Uniformly Continuous Semigroups 50
2.3. C0-Semigroups. General Properties 53
2.4. The Infinitesimal Generator 56
Problems 60
Notes 62
Chapter 3. Generation Theorems 63
3.1. The Hille-Yosida Theorem. Necessity 63
3.2. The Hille-Yosida Theorem. Sufficiency 66
3.3. The Feller-Miyadera-Phillips Theorem 68
3.4. The Lumer-Phillips Theorem 70
3.5. Some Consequences 73
3.6. Examples 75
3.7. The Dual of a C0-Semigroup 79
3.8. The Sun Dual of a C0-Semigroup 82
3.9. Stone Theorem 84
Problems 86
Notes 87
Chapter 4. Differential Operators Generating C0- Semigroups 89
4.1. The Laplace Operator with Dirichlet Boundary Condition 89
4.2. The Laplace Operator with Neumann Boundary Condition 95
4.3. The Maxwell Operator 96
4.4. The Directional Derivative 99
4.5. The Schrödinger Operator 102
4.6. The Wave Operator 103
4.7. The Airy Operator 107
4.8. The Equations of Linear Thermoelasticity 108
4.9. The Equations of Linear Viscoelasticity 110
Problems 113
Notes 115
Chapter 5. Approximation Problems and Applications 117
5.1. The Continuity of A .etA 117
5.2. The Chernoff and Lie-Trotter Formulae 122
5.3. A Perturbation Result 125
5.4. The Central Limit Theorem 126
5.5. Feynman Formula 129
5.6. The Mean Ergodic Theorem 133
Problems 138
Notes 139
Chapter 6. Some Special Classes of C0-Semigroups 141
6.1. Equicontinuous Semigroups 141
6.2. Compact Semigroups 145
6.3. Differentiable Semigroups 149
6.4. Semigroups with Symmetric Generators 156
6.5. The Linear Delay Equation 159
Problems 161
Notes 162
Chapter 7. Analytic Semigroups 163
7.1. Definition and Characterizations 163
7.2. The Heat Equation 168
7.3. The Stokes Equation 174
7.4. A Parabolic Problem with Dynamic Boundary Conditions 178
7.5. An Elliptic Problem with Dynamic Boundary Conditions 180
7.6. Fractional Powers of Closed Operators 182
7.7. Further Investigations in the Analytic Case 189
Problems 192
Notes 193
Chapter 8. The Nonhomogeneous Cauchy Problem 195
8.1. The Cauchy Problem u' = Au + f , u(a) =. 195
8.2. Smoothing Effect. The Hilbert Space Case 201
8.3. An Approximation Result 204
8.4. Compactness of the Solution Operator from LP(a, b X
8.5. The Case when ( .I – A) -1 is Compact 209
8.6. Compactness of the Solution Operator from L 1 (a, b X)
Problems 214
Notes 216
Chapter 9. Linear Evolution Problems with Measures as Data 217
9.1. The Problem du = {Au} dt + dg, u(a) =. 217
9.2. Regularity of £8 Solutions 222
9.3. A Characterization of £8 Solutions 225
9.4. Compactness of the £8 -Solution Operator 228
9.5. Evolution Equations with "Spatial" Measures as Data 232
Problems 235
Notes 237
Chapter 10. Some Nonlinear Cauchy Problems 239
10.1. Peano's Local Existence Theorem 239
10.2. The Problem u= f (t, u)+ g(t, u) 243
10.3. Saturated Solutions 248
10.4. The Klein-Gordon Equation 254
10.5. An Application to a Problem in Mechanics 257
Problems 259
Notes 260
Chapter 11. The Cauchy Problem for Semilinear Equations 261
11.1. The Problem u' = Au + f ( t , u) with f Lipschitz 261
11.2. The Problem u'= - Au + f (t, u) with f Continuous 265
11.3. Saturated Solutions 267
11.4. Asymptotic Behavior 273
11.5. The Klein-Gordon Equation Revisited 276
11.6. A Parabolic Semilinear Equation 277
Problems 279
Notes 280
Chapter 12. Semilinear Equations Involving Measures 281
12.1. The Problem du = {An} dt + dgu with u . gu Lipschitz 281
12.2. The Problem du - {An} dt + dgu with u . gu Continuous 285
12.3. Saturated £8-Solutions 288
12.4. The Case of Spatial Measures 294
12.5. Two Examples 296
12.6. One More Example 298
Problems 300
Notes 302
Appendix A. Compactness Results 303
A.1. Compact Operators 303
A.2. Compactness in C([a, b] X)
A.3. Compactness in C([a, b] Xw)
A.4. Compactness in LP(a, b X )
A.5. Compactness in LP(a, b X ) . Continued
A.6. The Superposition Operator 324
Problems 327
Notes 330
Solutions 331
Bibliography 373
List of Symbols 380
Subject Index 383