Nonlocal diffusion problems (Providence; Madrid, 2010). - ОГЛАВЛЕНИЕ / CONTENTS
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ОбложкаNonlocal diffusion problems / F.Andreu-Vaillo et al. - Providence: American Mathematical Society; Madrid: Real Sociedad Matemática Espaсola, 2010. - xv, 256 p.: ill. - (Mathematical surveys and monographs; vol.165). - Bibliogr.: p.249-254. - Ind.: p.255-256. - ISBN 978-0-8218-5230-9
 

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Оглавление / Contents
 
Preface ........................................................ xi

Chapter 1   The Cauchy problem for linear nonlocal diffusion .... 1
1.1  The Cauchy problem ......................................... 1
     1.1.1  Existence and uniqueness ............................ 5
     1.1.2  Asymptotic behaviour ................................ 6
1.2  Refined asymptotics ....................................... 10
     1.2.1  Estimates on the regular part of the fundamental
            solution ........................................... 12
     1.2.2  Asymptotics for the higher order terms ............. 17
     1.2.3  A different approach ............................... 20
1.3  Rescaling the kernel. A nonlocal approximation of the
     heat equation ............................................. 22
1.4  Higher order problems   ................................... 23
     1.4.1  Existence and uniqueness ........................... 24
     1.4.2  Asymptotic behaviour ............................... 25
     1.4.3  Rescaling the kernel in a higher order problem ..... 28
     Bibliographical notes ..................................... 29

Chapter 2   The Dirichlet problem for linear nonlocal
            diffusion .......................................... 31
2.1  The homogeneous Dirichlet problem ......................... 31
     2.1.1  Asymptotic behaviour  .............................. 32
2.2  The nonhomogeneous Dirichlet problem ...................... 36
     2.2.1  Existence, uniqueness and a comparison principle ... 36
     2.2.2  Convergence to the heat equation when rescaling
            the kernel ......................................... 38
     Bibliographical notes ..................................... 40

Chapter 3   The Neumann problem for linear nonlocal
            diffusion .......................................... 41
3.1  The homogeneous Neumann problem ........................... 41
     3.1.1  Asymptotic behaviour ............................... 42
3.2  The nonhomogeneous Neumann problem	........................ 45
     3.2.1  Existence and uniqueness ........................... 46
     3.2.2  Rescaling the kernels. Convergence to the heat
            equation ........................................... 48
     3.2.3  Uniform convergence in the homogeneous case ........ 54
     3.2.4  An L1 -convergence result in the nonhomogeneous
            case ............................................... 56
     3.2.5  A weak convergence result in the nonhomogeneous
            case ............................................... 57
     Bibliographical notes ..................................... 63

Chapter 4   A nonlocal convection diffusion problem ............ 65
4.1  A nonlocal model with a nonsymmetric kernel ............... 65
4.2  The linear semigroup revisited ............................ 69
4.3  Existence and uniqueness of the convection problem ........ 76
4.4  Rescaling the kernels. Convergence to the local
     convection-diffusion problem .............................. 82
4.5  Long time behaviour of the solutions ...................... 90
4.6  Weakly nonlinear behaviour ................................ 96
     Bibliographical notes ..................................... 98

Chapter 5   The Neumann problem for a nonlocal nonlinear
            diffusion equation ................................. 99
5.1  Existence and uniqueness of solutions .................... 100
     5.1.1  Notation and preliminaries ........................ 100
     5.1.2  Mild solutions and contraction principle .......... 104
     5.1.3  Existence of solutions ............................ 112
5.2  Rescaling the kernel. Convergence to the local problem ... 115
5.3  Asymptotic behaviour ..................................... 118
     Bibliographical notes .................................... 122

Chapter 6   Nonlocal p-Laplacian evolution problems ........... 123
6.1  The Neumann problem ...................................... 124
     6.1.1  Existence and uniqueness .......................... 125
     6.1.2  A precompactness result ........................... 128
     6.1.3  Rescaling the kernel. Convergence to the local
            p-Laplacian ....................................... 131
     6.1.4  A Poincare type inequality ........................ 137
     6.1.5  Asymptotic behaviour .............................. 141
6.2  The Dirichlet problem .................................... 142
     6.2.1  A Poincare type inequality ........................ 144
     6.2.2  Existence and uniqueness of solutions ............. 146
     6.2.3  Convergence to the local p-Laplacian .............. 149
     6.2.4  Asymptotic behaviour .............................. 153
6.3  The Cauchy problem ....................................... 154
     6.3.1  Existence and uniqueness .......................... 154
     6.3.2  Convergence to the Cauchy problem for the local
            p-Laplacian ....................................... 157
6.4  Nonhomogeneous problems .................................. 160
     Bibliographical notes .................................... 161

Chapter 7   The nonlocal total variation flow ................. 163
7.1  Notation and preliminaries ............................... 164
7.2  The Neumann problem ...................................... 165
     7.2.1  Existence and uniqueness .......................... 166
     7.2.2  Rescaling the kernel. Convergence to the total
            variation flow .................................... 169
     7.2.3  Asymptotic behaviour .............................. 174
7.3  The Dirichlet problem .................................... 175
     7.3.1  Existence and uniqueness .......................... 176
     7.3.2  Convergence to the total variation flow ........... 180
     7.3.3  Asymptotic behaviour .............................. 188
     Bibliographical notes .................................... 189

Chapter 8   Nonlocal models for sandpiles ..................... 191
8.1  A nonlocal version of the Aronsson-Evans-Wu model for
     sandpiles ................................................ 191
     8.1.1  The Aronsson-Evans-Wu model for sandpiles ......... 191
     8.1.2  Limit as p → ∞ in the nonlocal p-Laplacian
            Cauchy problem .................................... 193
     8.1.3  Rescaling the kernel. Convergence to the local
            problem ........................................... 195
     8.1.4  Collapse of the initial condition ................. 197
     8.1.5  Explicit solutions ................................ 200
     8.1.6  A mass transport interpretation ................... 210
     8.1.7  Neumann boundary conditions ....................... 212
8.2  A nonlocal version of the Prigozhin model for
     sandpiles ................................................ 213
     8.2.1  The Prigozhin model for sandpiles ................. 214
     8.2.2  Limit as p → ∞ in the nonlocal p-Laplacian
            Dirichlet problem ................................. 214
     8.2.3  Convergence to the Prigozhin model ................ 217
     8.2.4  Explicit solutions ................................ 219

Bibliographical notes ......................................... 222

Appendix A.  Nonlinear semigroups ............................. 223
A.l  Introduction ............................................. 223
A.2  Abstract Cauchy problems ................................. 224
A.3  Mild solutions ........................................... 227
A.4  Accretive operators ...................................... 229
A.5  Existence and uniqueness theorem ......................... 235
A.6  Regularity of the mild solution .......................... 239
A.7  Convergence of operators ................................. 241
A.8  Completely accretive operators ........................... 242

Bibliography .................................................. 249
Index ......................................................... 255


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