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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