This volume includes several invited lectures given at the International Workshop "Analysis, Partial Differential Equations and Applications", held at the Mathematical Department of Sapienza University of Rome, on the occasion of the 70th birthday of Vladimir G. Maz'ya, a renowned mathematician and one of the main experts in the field of pure and applied analysis. The book aims at spreading the seminal ideas of Maz'ya to a larger audience in faculties of sciences and engineering. In fact, all articles were inspired by previous works of Maz'ya in several frameworks, including classical and contemporary problems connected with boundary and initial value problems for elliptic, hyperbolic and parabolic operators, Schr?dinger-type equations, mathematical theory of elasticity, potential theory, capacity, singular integral operators, p-Laplacians, functional analysis, and approximation theory. Maz'ya is author of more than 450 papers and 20 books. In his long career he obtained many astonishing and frequently cited results in the theory of harmonic potentials on non-smooth domains, potential and capacity theories, spaces of functions with bounded variation, maximum principle for higher-order elliptic equations, Sobolev multipliers, approximate approximations, etc. The topics included in this volume will be particularly useful to all researchers who are interested in achieving a deeper understanding of the large expertise of Vladimir Maz'ya.
Author(s): Alberto Cialdea, Flavia Lanzara, Paolo Emilio Ricci
Edition: 1
Year: 2009
Language: English
Pages: 342
Cover......Page 1
Editors/volume Information......Page 3
Title......Page 4
Copyright......Page 5
Contents......Page 6
Photo: Vladimir Maz’ya......Page 8
On the Occasion of the 70th Birthday of Vladimir Maz’ya......Page 9
References......Page 13
Books by Vladimir Maz’ya......Page 15
Books and articles in honor of Vladimir Maz’ya......Page 17
1. Introduction......Page 18
Notations.......Page 19
2. One-sided densities......Page 20
3. Boundary trace......Page 23
5. Summability of traces......Page 26
7. Embedding theorems......Page 27
8. The Gauss-Green formula......Page 28
9. Average trace of BV (Ω) functions......Page 29
References......Page 30
1. Introduction......Page 31
2. Notation and conventions......Page 32
4. Our model......Page 33
5. Choosing a common language......Page 35
6. Special case with no dependence on x³......Page 36
7. Discussion......Page 43
References......Page 45
1. Introduction......Page 46
2. Local and global Hölder estimates for subsolutions......Page 48
3. Lipschitz estimates......Page 51
4. The Dirichlet problem......Page 53
References......Page 54
1. Introduction......Page 56
2. A basic lemma and some consequences......Page 57
3. The main result......Page 59
4. The constant coefficients case......Page 61
5. Smooth coefficients......Page 62
6. The two-dimensional elasticity......Page 64
7. Systems of ordinary differential equations......Page 65
8. Systems of partial differential equations......Page 68
References......Page 69
1. Introduction......Page 72
2. Estimates for potentials......Page 73
3. Applications to PDE’s......Page 76
References......Page 78
Abstract......Page 80
1. Introduction......Page 81
2. Necessary conditions for the discrete model with positive data......Page 83
3. Sufficient conditions for the discrete model......Page 87
4. Summation by parts lemmas......Page 92
References......Page 94
1. Introduction......Page 96
2. Slowly oscillating data......Page 98
3. Mellin pseudodifferential operators......Page 100
4. Fredholm theory for the algebra Up......Page 102
5. Applications of Mellin pseudodifferential operators......Page 103
6. Banach algebra G of generalized singular integral operators. Fredholmness and index......Page 105
7. Generalized singular integral operators with a Carleman shift......Page 106
References......Page 109
1. Introduction......Page 111
2. The heat equation......Page 112
3. Nonuniqueness in the direct problem......Page 113
4. Uniqueness in the direct problem......Page 114
7. Exponential solutions......Page 115
8. Generalized trigonometric polynomials......Page 118
9. Finding the coefficients of a past temperature function......Page 119
10. Finding the coefficients of a memory function......Page 120
11. An isometry......Page 121
13. The inverse problem......Page 122
14. Other norms for the space of past temperature functions......Page 123
15. Other norms for the space of memory functions......Page 125
16. What remains to be done?......Page 127
References......Page 128
1. Introduction......Page 129
2. Pointwise estimates for the gradient of bounded or semibounded
harmonic functions in multidimensional domains......Page 131
3. Sharp pointwise estimates for the gradient of harmonic
functions in the half-space with boundary data from Lp......Page 134
Acknowledgment......Page 141
References......Page 142
1. Introduction......Page 143
2. Quasi-interpolation on uniform grids......Page 144
3.1. Quasi-interpolation with perturbed uniform grid......Page 148
3.2. Quasi-interpolation with scattered grids......Page 150
4. Application to the computation of integral operators......Page 152
References......Page 155
1. Introduction......Page 157
2. Preliminaries......Page 159
3. Estimates for the Green function......Page 160
4. Applications: estimates on solutions of the Dirichlet problem......Page 168
References......Page 171
1. Introduction......Page 173
2. Discussion of the proofs......Page 179
References......Page 183
1. Introduction......Page 184
2. Analysis in uniformly rectifiable domains......Page 187
3. The harmonic measure......Page 191
4. The Green function......Page 195
5. Proofs of main results......Page 198
6. Appendix......Page 206
References......Page 207
1. Introduction......Page 209
2.1 Operators on lattices and discrete groups......Page 212
2.2 Operators on quantum graphs......Page 214
3.1 Discrete case......Page 217
3.2 Continuous case......Page 219
4.1 Free groups......Page 220
4.2 General remark on left invariant diffusions on Lie groups......Page 222
4.3 Heisenberg group......Page 223
4.4 Group Aff ( R¹) of affine transformations of the real line......Page 224
References......Page 225
1. Introduction......Page 227
2. Preliminaries......Page 228
3. Properties of ACL-homeomorphisms......Page 230
4. ACL-homeomorphisms and first-order systems of PDE’s......Page 233
5. Mappings with integrable distortion......Page 235
References......Page 236
1. Introduction......Page 238
2.1. Geometrical description of the composite configuration.......Page 239
2.2. Field equations in Ω(m)......Page 240
2.3. Field equations in Ω......Page 242
2.4. Formulation of the interface crack problem......Page 243
2.5. Green’s formulas and uniqueness theorem......Page 244
3.1. Layer potentials......Page 245
3.2. Reduction to boundary equations......Page 246
3.3. Existence theorems and regularity of solutions......Page 248
References......Page 253
1. Introduction and preliminaries......Page 255
2. Positive Lagrangian representations......Page 258
3. Coercivity and ground state......Page 259
4. Ground states and minimal growth at infinity......Page 262
5. Liouville theorems......Page 263
6. Variational principle for solutions of minimal growth and
comparison principle......Page 265
7. Solvability of nonhomogeneous equation......Page 266
8. Criticality theory......Page 268
9. The linear case (p =2)......Page 271
9.1. Liouville-type theorem......Page 272
9.2. The space D1,2
A,V......Page 273
9.3. Positive solutions of minimal growth......Page 274
References......Page 275
1. Introduction......Page 278
2. Mixed boundary value problems for the Stokes system......Page 280
Operator pencils generated by the boundary value proble......Page 281
Regularity assertions in weighted Sobolev spaces......Page 283
The maximum principle......Page 284
Regularity assertions for variational solutions......Page 285
Existence of solutions in W1,s (G )× Ls (G)......Page 286
References......Page 287
1. Introduction......Page 290
1.1. Typical examples of operators with variable orders......Page 291
1.2. Typical examples of spaces with variable exponents......Page 292
2. Some basics for variable exponent Lebesgue spaces......Page 293
3. On some recent results on boundedness of classical operators
in spaces Lp(·)
(Ω,)......Page 294
3.2. On the maximal operator......Page 295
3.3. On the Cauchy singular operator......Page 296
3.4. On potential operators......Page 297
4. Maximal and potential operators in variable exponent Morrey spaces......Page 300
5. Fractional integrals and hypersingular integrals in variable order Hölder spaces on homogeneous spaces......Page 301
5.2. Theorems on mapping properties......Page 304
References......Page 305
1. Introduction......Page 311
2. The von Koch curve......Page 313
3. Koch mixtures......Page 317
4. Sierpiński gasket......Page 321
References......Page 325
1. Introduction......Page 327
2. The Laplacian......Page 328
3.1. Continuous boundary data......Page 330
3.2. Square integrable boundary data......Page 332
4.1. Continuous boundary data......Page 333
4.2. Square integrable boundary data......Page 336
4.3. Data in boundary energy spaces......Page 337
References......Page 338
