0-444-51743-X, 0-444-52846-6, 978-0-444-52846-9
Table of contents :
Y……Page
Preface v 6……Page 0006
— 1,p(x)……Page 0016
p(x)-Laplacian, 3……Page 0018
image recovery, 6……Page 0021
space, generalized, 10……Page 0025
— Lp(x)(+), 12……Page 0027
— inequality, 14……Page 0029
embedding theorems, 15……Page 0030
Young inequality, 16……Page 0031
3.2. Generalized diffusion equation 24……Page 0039
3.3. Equationswith convection terms 32……Page 0047
— eigenvalue problem for, 34……Page 0049
4.1. Uniqueness of solution of the generalized p(x)-Laplace equation 36……Page 0051
4.2. Uniqueness of solution of the generalized diffusion equation 40……Page 0055
local weak solutions, 43……Page 0058
— relation, 47……Page 0062
5.3. The ordinary differential inequality 49……Page 0064
5.4. Equationswith convection terms 55……Page 0070
directional localization, 56……Page 0071
6.2. Generalized p(x)-Laplace equation 68……Page 0083
unbounded domains, 72……Page 0087
8. Systems of elliptic equations 76……Page 0091
8.2. Localization properties 78……Page 0093
8.3. Systems of other types 81……Page 0096
borderline cases, 82……Page 0097
9.2. The ordinary differential inequality in the limit case 95……Page 0110
References 97……Page 0112
CHAPTER 2. A HANDBOOK OF ? -CONVERGENCE 101……Page 0116
1. Introduction 103……Page 0118
Notation 109……Page 0124
2.1. The definitions of ? -convergence 110……Page 0125
2.2. ? -convergence and lower semicontinuity 113……Page 0128
relaxation, 114……Page 0129
equicoercive sequence, 115……Page 0130
3. Localization methods 117……Page 0132
3.1. Supremumofmeasures 118……Page 0133
— technique, 119……Page 0134
3.3. Ageneral compactness procedure 120……Page 0135
slicing method, 122……Page 0137
4.1. Aprototypical compactness theorem 124……Page 0139
quasiconvex envelope, 128……Page 0143
4.3. Convergence of quadratic forms 131……Page 0146
G-convergence, 132……Page 0147
— theory, 134……Page 0149
homogenized functional, 135……Page 0150
cell-problem homogenization formula, 137……Page 0152
5.3. Homogenization of quadratic forms 138……Page 0153
— principle, 140……Page 0155
closure of Riemannian metrics, 142……Page 0157
perforated domain, 144……Page 0159
relaxed Dirichlet problem, 149……Page 0164
obstacle problem, 151……Page 0166
6.4. Double-porosity homogenization 153……Page 0168
perimeter, 155……Page 0170
elastica functional, 158……Page 0173
7.3. Acompactness result 164……Page 0179
7.4. Other functionals generating phase-transitions 165……Page 0180
Gibbs phenomenon, 171……Page 0186
Ginzburg–Landau energy, 173……Page 0188
— embedding, 176……Page 0191
9.1. TheLeDret–Raoult result 180……Page 0195
9.2. Acompactness theorem 183……Page 0198
9.3. Higher-order ? -limits 185……Page 0200
isometric embedding, 186……Page 0201
10.1. Special functions with bounded variation 187……Page 0202
10.2. TheAmbrosio–Tortorelli approximation 190……Page 0205
10.3. Other approximations 193……Page 0208
finite-difference energies, 196……Page 0211
pair potential, 199……Page 0214
nearest neighbor, 200……Page 0215
ferromagnetic interaction, 205……Page 0220
References 208……Page 0223
CHAPTER 3. BUBBLING IN NONLINEAR ELLIPTIC PROBLEMS NEAR CRITICALITY 215……Page 0230
— elliptic boundary value problems, 217……Page 0232
2.1. Ansatz and scheme of the proof 225……Page 0240
2.2. Variational reduction and conclusion of the proof 233……Page 0248
3.1. The case of a small hole 255……Page 0270
3.2. Bubbling under symmetries 266……Page 0281
4. The Brezis–Nirenberg problem in dimension N =3: the proof ofTheorem1.2 269……Page 0284
4.1. Energy expansion of single bubbling 270……Page 0285
4.2. Themethod of proof 273……Page 0288
4.3. The linear problem 276……Page 0291
4.4. Solving the nonlinear problem 277……Page 0292
4.5. Variational formulation of the reduced problem for k =1 278……Page 0293
4.6. Proof ofTheorem1.2, part (a): single bubbling 279……Page 0294
4.7. Proof of Theorem 1.2, part (b): multiple bubbling 280……Page 0295
5.1. Proof ofTheorem1.3 285……Page 0300
5.2. A related 2-d problem involving nonlinearity with large exponent 309……Page 0324
References 312……Page 0327
CHAPTER 4. SINGULAR ELLIPTIC AND PARABOLIC EQUATIONS 317……Page 0332
1. Introduction 319……Page 0334
spectrum of singular eigenvalue problem, 325……Page 0340
monotone nonlinearities, 333……Page 0348
4. Existence of solutions via sub- and supersolutions 337……Page 0352
sublinear, 340……Page 0355
behavior of solutions, 341……Page 0356
— differenciability of, 350……Page 0365
linearized stability, 353……Page 0368
stabilization, 358……Page 0373
10.1. Power law: Lu = ?K(x)uq 362……Page 0377
10.2. First combination of two power laws: Lu+M(x)up = ?K(x)uq 364……Page 0379
10.3. Second combination of power laws: Lu = ?K(x)uq + M(x)up 367……Page 0382
— variational methods, 370……Page 0385
12. Results for the radial and the one-dimensional problem 376……Page 0391
— solutions, 383……Page 0398
Acknowledgements 388……Page 0403
CHAPTER 5. SCHAUDER-TYPE ESTIMATES AND APPLICATIONS 401……Page 0416
1.1. What areSchauder-type estimates? 403……Page 0418
1.2. Why dowe needSchauder estimates? 404……Page 0419
1.3. Why so many methods of proof? 405……Page 0420
1.4. Classification of proofs 406……Page 0421
Euler–Poisson–Darboux, 407……Page 0422
— transform, 408……Page 0423
— balayage method, 410……Page 0425
— spaces, 411……Page 0426
Littlewood–Paley (LP), 412……Page 0427
weighted norms, 413……Page 0428
interpolation inequalities, 415……Page 0430
— curvature, 416……Page 0431
2.6. Integral characterization ofHolder continuity 418……Page 0433
3.1. Direct arguments frompotential theory 419……Page 0434
3.2. C1+? estimates via themaximumprinciple 423……Page 0438
3.3. C2+? estimates via Littlewood–Paley theory 425……Page 0440
— inequality, 426……Page 0441
3.5. Other methods 430……Page 0445
tangential operator, 432……Page 0447
Schwarz reflection principle, 434……Page 0449
Poincare–Perron, 435……Page 0450
5.1. First “type (I)” result 436……Page 0451
5.2. Second “type (I)” result 438……Page 0453
method of continuity, 440……Page 0455
Schauder fixed-point theorem, 441……Page 0456
6.3. Fixed-point theory and theDirichlet problem 445……Page 0460
Krein–Rutman theorem, 446……Page 0461
6.5. Method of sub- and supersolutions 448……Page 0463
6.6. Asymptotics near isolated singularities or at infinity 449……Page 0464
Loewner–Nirenberg equation, 451……Page 0466
6.8. First comparison argument 454……Page 0469
References 461……Page 0476
A Appendix A. Proof of Proposition 2.3 389……Page 0404
B Appendix B. A strong maximum principle for second-order equations with locally bounded coefficients . 393……Page 0408
References 395……Page 0410
CHAPTER 6. THE DAM PROBLEM 465……Page 0480
— inequalities, 467……Page 0482
The weak formulation of Alt and Brezis–Kinderlehrer–Stampacchia 468……Page 0483
Outline of the chapter 469……Page 0484
Notation 470……Page 0485
1.1. Formulation of the problem 471……Page 0486
— unified, 475……Page 0490
monotonicity property, 486……Page 0501
2.1. Some properties of the solutions 491……Page 0506
2.2. Continuity of the free boundary 495……Page 0510
2.3. Existence and uniqueness of minimal and maximal solutions 498……Page 0513
S3-connected solution, 507……Page 0522
2.5. Uniqueness of the reservoirs-connected solution 510……Page 0525
3.1. Properties of the solutions 516……Page 0531
3.2. Continuity of the free boundary 526……Page 0541
3.3. Existence and uniqueness of minimal and maximal solutions 533……Page 0548
3.4. Reservoirs-connected solution 544……Page 0559
3.5. Uniqueness of the reservoirs-connected solution 545……Page 0560
References 551……Page 0566
CHAPTER 7. NONLINEAR EIGENVALUE PROBLEMS FOR HIGHER-ORDER MODEL EQUATIONS 553……Page 0568
Fisher–Kolmogorov equation, extended, 555……Page 0570
2. Decreasing nonlinearity f (u) 560……Page 0575
2.1. Existence of a periodic solution 561……Page 0576
2.2. Uniqueness 565……Page 0580
2.3. Asymptotics 569……Page 0584
3. Asuperlinear bifurcation problem 573……Page 0588
resonance, 574……Page 0589
3.2. Multibump periodic solutions 576……Page 0591
3.3. Branches of periodic solutions 583……Page 0598
4. Asublinear bifurcation problem 585……Page 0600
— 593……Page 0608
5.1. Multibump solutions 594……Page 0609
5.2. Branches of multibump periodic solutions 599……Page 0614
References 603……Page 0618
gamma function, 601……Page 0616
Subject Index 609……Page 0624
V O L U M E I – Short Link……Page HB-PDE_Vol.1.djvu
Preface v……Page HB-PDE_Vol.1.djvu#6
CHAPTER 1. Solutions of Quasilinear Second-Order Elliptic Boundary Value Problems via Degree Theory……Page HB-PDE_Vol.1.djvu#12
CHAPTER 2. Stationary Navier–Stokes Problem in a Two-Dimensional Exterior Domain……Page HB-PDE_Vol.1.djvu#82
CHAPTER 3. Qualitative Properties of Solutions to Elliptic Problems……Page HB-PDE_Vol.1.djvu#168
CHAPTER 4. On Some Basic Aspects of the Relationship between the Calculus of Variations and Differential Equations……Page HB-PDE_Vol.1.djvu#246
CHAPTER 5. On a Class of Singular Perturbation Problems……Page HB-PDE_Vol.1.djvu#308
CHAPTER 6. Nonlinear Spectral Problems for Degenerate Elliptic Operators……Page HB-PDE_Vol.1.djvu#396
CHAPTER 7. Analytical Aspects of Liouville-Type Equations with Singular Sources……Page HB-PDE_Vol.1.djvu#502
CHAPTER 8. Elliptic Equations Involving Measures……Page HB-PDE_Vol.1.djvu#604
Author Index ……Page HB-PDE_Vol.1.djvu#724
Subject Index ……Page HB-PDE_Vol.1.djvu#732
V O L U M E II – Short Link……Page HB-PDE_Vol.2.djvu
Preface v……Page HB-PDE_Vol.2.djvu#6
Contents of Volume I xi……Page HB-PDE_Vol.2.djvu#12
CHAPTER 1. The Dirichlet Problem for Superlinear Elliptic Equations……Page HB-PDE_Vol.2.djvu#14
CHAPTER 2. Nonconvex Problems of the Calculus of Variations and Differential Inclusions……Page HB-PDE_Vol.2.djvu#70
CHAPTER 3. Bifurcation and Related Topics in Elliptic Problems……Page HB-PDE_Vol.2.djvu#140
CHAPTER 4. Metasolutions: Malthus versus Verhulst in Population Dynamics. A Dream of Volterra……Page HB-PDE_Vol.2.djvu#224
CHAPTER 5. Elliptic Problems with Nonlinear Boundary Conditions and the Sobolev Trace Theorem……Page HB-PDE_Vol.2.djvu#324
CHAPTER 6. Schrodinger Operators with Singular Potentials……Page HB-PDE_Vol.2.djvu#420
CHAPTER 7. Multiplicity Techniques for Problems without Compactness……Page HB-PDE_Vol.2.djvu#532
Author Index ……Page HB-PDE_Vol.2.djvu#614
Subject Index ……Page HB-PDE_Vol.2.djvu#622
ПРЕДМЕТНЫЙ УКАЗАТЕЛЬ(eng)……Page 1
A-harmonic, 488……Page 503
512……Page 527
515……Page 530
Ambrosio–Tortorelli energies, 191……Page 0206
nonhomogeneous, 9……Page 0024
anisotropy, essential, 73……Page 0088
Blake–Zisserman approximation, 206……Page 0221
Minkowski content, 188……Page 0203
baseline solution, 559……Page 574
594……Page 609
beta function, 600……Page 0615
— structure, 556……Page 0571
— curves, 559……Page 0574
branches, 573……Page 588
585……Page 600
— of positive solutions, 354……Page 0369
free boundary, 471……Page 486
491……Page 506
— leaky, 469……Page 484
516……Page 531
weak formulation, 473……Page 0488
set of finite perimeter, 156……Page 0171
582……Page 597
583……Page 598
599……Page 614
600……Page 615
critical exponent, 218……Page 233
224……Page 239
subadditivity, 157……Page 0172
Robin function, 179……Page 0194
capacity, 145……Page 0160
16……Page 31
24……Page 39
next-to-nearest neighbor, 203……Page 0218
— operator, 444……Page 0459
Orlicz–Sobolev space, 11……Page 0026
compression of a cone, 447……Page 0462
— terms, 32……Page 47
55……Page 70
— functionals, 194……Page 0209
220……Page 235
223……Page 238
— generalized, 9……Page 24
— uniqueness of, 510……Page 525
545……Page 560
— nonlinear, 468……Page 483
511……Page 526
slow diffusion–strong absorption, 5……Page 0020
— form, 133……Page 0148
Poisson equation, 409……Page 0424
electrorheological fluids, 7……Page 0022
Schrodinger equation, 556……Page 571
— identity, 560……Page 575
initial value problem, 562……Page 577
572……Page 587
— functions, 44……Page 59
79……Page 94
570……Page 585
571……Page 586
577……Page 592
595……Page 610
— of anisotropic diffusion, 86……Page 0101
— of mixed type, 90……Page 0105
— with convective terms, 88……Page 0103
fixed-point theorems, 443……Page 0458
Jacobian, distributional, 175……Page 0190
490……Page 505
495……Page 510
526……Page 541
renormalized unknown, 452……Page 0467
— equation, 3……Page 18
SBV compactness theorem, 189……Page 0204
— inequality, 13……Page 0028
— inverse, 38……Page 0053
Lax formula, 143……Page 0158
— of networks, 204……Page 0219
implicit function theorem, 563……Page 0578
576……Page 591
Ising system, 166……Page 181
205……Page 220
449……Page 464
450……Page 465
laminate, 139……Page 0154
557……Page 572
points of symmetry, 561……Page 576
584……Page 599
separation of scales, 207……Page 0222
recovery sequence, 111……Page 0126
line-tension effect, 168……Page 0183
— property, 450……Page 0465
upper bound, 104……Page 0119
log-continuous, 17……Page 0032
minimal solution, 337……Page 352
451……Page 466
502……Page 517
539……Page 554
maximum principle, 393……Page 408
423……Page 438
589……Page 604
590……Page 605
mean curvature, 416……Page 431
455……Page 470
Modica–Mortola theorem, 159……Page 0174
Moreau–Yosida transform, 112……Page 0127
uniqueness of principal eigenvalue, 332……Page 0347
558……Page 573
593……Page 608
Newtonian potential, 408……Page 423
419……Page 434
7……Page 22
8……Page 23
nonoscillation result, 494……Page 509
520……Page 535
nonstandard, 4……Page 0019
— growth condition, 4……Page 19
viscous fluid, 8……Page 0023
optimal profile, 162……Page 0177
407……Page 422
586……Page 601
quasiconvexity, 127……Page 0142
pool, 509……Page 0524
— eigenvalue, 329……Page 344
330……Page 345
10……Page 25
— anisotropic, 89……Page 0104
— uniqueness of, 377……Page 0392
reservoirs-connected solution, 507……Page 522
544……Page 559
549……Page 564
— equation with nonlinear absorption term, 83……Page 0098
shooting argument, 562……Page 0577
— elliptic, 323……Page 0338
— estimates for, 327……Page 342
334……Page 349
— estimates for eigenvalues, 329……Page 0344
wetting condition, 172……Page 0187
— 0……Page 15
12……Page 27
— comparison principle, 489……Page 0504
— maximum principle, 11……Page 26
sub- and supersolutions, 410……Page 425
448……Page 463
supersolution, 460……Page 0475
Swift–Hohenberg equation, 555……Page 570
thin film, 181……Page 0196
trace-interpolation inequality, 50……Page 0065
vanishing absorption, 92……Page 0107
— problem, 477……Page 0492
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