Sobolev spaces become the established and universal language of partial differential equations and mathematical analysis. Among a huge variety of problems where Sobolev spaces are used, the following important topics are the focus of this volume: boundary value problems in domains with singularities, higher order partial differential equations, local polynomial approximations, inequalities in Sobolev-Lorentz spaces, function spaces in cellular domains, the spectrum of a Schrodinger operator with negative potential and other spectral problems, criteria for the complete integration of systems of differential equations with applications to differential geometry, some aspects of differential forms on Riemannian manifolds related to Sobolev inequalities, Brownian motion on a Cartan-Hadamard manifold, etc. Two short biographical articles on the works of Sobolev in the 1930s and the foundation of Akademgorodok in Siberia, supplied with unique archive photos of S. Sobolev are included.
Author(s): Vladimir Maz'ya
Edition: 1
Year: 2008
Language: English
Pages: 390
0387856498......Page 1
Contents......Page 21
On the Mathematical Works of S.L. Sobolev in the 1930s......Page 27
References......Page 34
Sobolev in Siberia......Page 36
1. Introduction......Page 43
2. Boundary Harnack Principle and Carleson Estimate in Terms of the Green Function......Page 47
3. Proof of the Main Result......Page 48
References......Page 54
Sobolev Spaces and their Relatives: Local Polynomial Approximation Approach......Page 55
1. Topics in Local Polynomial Approximation Theory......Page 56
2. Local Approximation Spaces......Page 67
2.1 Λ-spaces......Page 72
2.2 M-spaces......Page 75
2.3 T -spaces......Page 77
2.4 V-spaces......Page 78
3.1 Embeddings......Page 81
3.2 Extensions......Page 83
3.3 Pointwise differentiability......Page 86
3.4 Nonlinear Approximation......Page 89
References......Page 91
1. Introduction......Page 93
2. Preliminaries and Notation......Page 96
3. Open Sets with Continuous Boundaries......Page 101
4. The Case of Diffeomorphic Open Sets......Page 103
5. Estimates for Dirichlet Eigenvalues via the Atlas Distance......Page 106
6. Estimates for Neumann Eigenvalues via the Atlas Distance......Page 110
8.1 On the atlas distance......Page 122
8.2 Comparison of atlas distance, Hausdorff–Pompeiu distance, and lower Hausdorff–Pompeiu deviation......Page 124
References......Page 125
1. Introduction......Page 127
2. Preliminaries......Page 129
3. Sobolev–Lorentz p, q-Capacitance......Page 133
4. Conductor Inequalities......Page 136
5. Necessary and Sufficient Conditions for Two-Weight Embeddings......Page 139
References......Page 143
Besov Regularity for the Poisson Equation in Smooth and Polyhedral Cones......Page 146
2. Regularity Result for a Smooth Cone......Page 149
3. Besov Regularity for the Neumann Problem......Page 157
4. Appendix A. Regularity of Solutions of the Poisson Equation......Page 162
5.2 Sobolev spaces on domains......Page 164
5.3 Besov spaces and wavelet......Page 165
References......Page 167
Variational Approach to Complicated Similarity Solutions of Higher Order Nonlinear Evolution Partial Differential Equations.......Page 169
1.1 Three types of nonlinear PDEs under consideration......Page 170
1.2 (I) Combustion type models: regional blow-up, global stability, main goals, and first discussion......Page 171
1.3 (II) Regional blow-up in quasilinear hyperbolic equations......Page 173
1.4 (III) Nonlinear dispersion equations and compactons......Page 174
2.1 Global existence and blow-up in higher order parabolic equations......Page 176
2.2 Blow-up data for higher order parabolic and hyperbolic PDEs......Page 182
2.3 Blow-up rescaled equation as a gradient system: towards the generic blow-up behavior for parabolic PDEs......Page 183
3.1 Variational setting and compactly supported solutions......Page 184
3.2 The Luste.rnik–Schnirelman theory and direct application of fibering method......Page 185
3.3 On a model with an explicit description of the Lusternik–Schnirelman sequence......Page 187
3.4 Preliminary analysis of geometric shapes of patterns......Page 188
4. Oscillation Problem: Local Oscillatory Structure of Solutions Close to Interfaces and Periodic Connections with Singularities......Page 195
4.1 Autonomous ODEs for oscillatory components......Page 196
4.2 Periodic oscillatory components......Page 197
4.3 Numerical construction of periodic orbits; m = 2......Page 198
4.4 Numerical construction of periodic orbits; m = 3......Page 199
5. Numeric Problem: Numerical Construction and First Classification of Basic Types of Localized Blow-up or Compacton Patterns......Page 203
5.1 Fourth order equation: m = 2......Page 204
5.2 Countable family of {F[sub(0)], F[sub(0)]}-interactions......Page 207
5.3 Countable family of {–F[sub(0)], F[sub(0)]}-interactions......Page 210
5.4 Periodic solutions in R......Page 211
5.6 More complicated patterns: towards chaotic structures......Page 213
References......Page 217
1.1 L[sub(q,p)]-cohomology and Sobolev inequalities......Page 220
1.2 Statement of the main result......Page 222
2. Manifolds with Contraction onto the Closed Unit Ball......Page 224
3. Proof of the Main Result......Page 226
References......Page 228
1. Introduction......Page 230
2. Heat Equation Solution Estimates......Page 234
3. Escape Rate of Brownian Motion......Page 237
4.2 General model manifolds......Page 241
References......Page 245
1. Introduction......Page 247
2. Preliminaries......Page 252
3. Smoothness Spaces on Lipschitz Boundaries and Lipschitz Domains......Page 254
4. The Case of C[sup(2)] Domains......Page 263
5. Approximation Scheme......Page 266
6. Proof of Step I......Page 269
7. Proof of Step II......Page 276
8. Proof of Step III......Page 278
References......Page 279
1.1 Introduction......Page 281
1.2 Spectral boundary value problem......Page 282
1.3 Polynomial property and the Korn inequality......Page 285
1.4 Formulation of the problem in the operator form......Page 287
1.5 Contents of the paper......Page 288
2.1 Model problem in the quasicylinder......Page 290
2.2 The Fredholm property of the problem operator......Page 292
2.3 Exponential decay and finite dimension of the kernel......Page 296
2.4 Continuous spectrum......Page 299
2.5 On the positive threshold......Page 304
3.1 The absence of the point spectrum......Page 305
3.2 Concentration of the discrete spectrum......Page 307
3.3 Comparison principles......Page 310
3.4 Artificial boundary conditions......Page 311
3.5 Opening gaps in the continuous spectrum......Page 315
3.6 Variational methods for searching trapped modes below the cut-off......Page 319
3.7 Remarks on cracks and edges......Page 323
3.8 Piezoelectric bodies......Page 324
References......Page 326
1. Introduction......Page 330
2. Notation and Preliminaries......Page 331
3. Remarks on Completely Integrable Linear Systems of Differential Equations......Page 333
4. Estimates for Operators Satisfying the Complete Integrability Condition......Page 337
5. Stability of Solutions of Completely Linear Integrable Systems......Page 341
6. Applications to Differential Geometry......Page 344
References......Page 346
1. Introduction......Page 347
2.1 The Rozenblum–Lieb–Cwikel estimate......Page 350
3.1 The approach by Li and Yau......Page 351
3.2 The approach by Lieb......Page 352
4. Operators on R[sup(d)], d ≥ 3: Non-Semiclassical Behavior of N_(0;H[sub(αV)]).......Page 354
5.1 Semiclassical behavior......Page 356
5.2 Non-semiclassical behavior of N_(0;H[sub(αv)]......Page 358
6.1 Semiclassical behavior......Page 359
7.1 Preliminary remarks......Page 361
7.2 Hyperbolic Laplacian......Page 363
8. Operators on Manifolds: Beyond Theorem 3.4......Page 364
9 Schrödinger Operator on a Lattice......Page 367
References......Page 370
1.1 Introduction......Page 372
1.2 Definitions......Page 373
1.3 Wavelet systems and sequence spaces......Page 376
1.4 Domains......Page 378
1.5 Some properties......Page 379
1.6 Frames......Page 382
1.7 Bases......Page 384
2.1 Traces and extensions......Page 385
2.2 Approximation, density, decomposition......Page 389
3. Spaces on Cubes and Polyhedrons......Page 393
4.1 Cubes and polyhedrons......Page 397
4.3 Comments......Page 399
References......Page 401
N......Page 403
W......Page 404
