Sobolev spaces: with applications to elliptic partial differential equations
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| Format: | Book |
| Language: | English Russian |
| Published: |
Heidelberg ; Dordrecht ; London ; New York
Springer
[2011]
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| Edition: | 2nd, revised and augmented edition |
| Series: | Grundlehren der mathematischen Wissenschaften
342 |
| Subjects: | |
| Online Access: | Inhaltstext Inhaltsverzeichnis |
| Item Description: | The 1st edition, published in 1985 in English under Vladimir G. Maz’ja in the Springer Series of Soviet Mathematics was translated from Russian by Tatyana O. Shaposhnikova Aus dem Russischen übersetzt |
| Physical Description: | xxviii, 866 Seiten Illustrationen, Diagramme |
| ISBN: | 9783642155635 9783662517291 |
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MARC
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| 084 | |a 510 |2 sdnb | ||
| 100 | 1 | |a Mazʹja, Vladimir Gilelevič |d 1937- |e Verfasser |0 (DE-588)121490602 |4 aut | |
| 245 | 1 | 0 | |a Sobolev spaces |b with applications to elliptic partial differential equations |c Vladimir Maz'ya |
| 250 | |a 2nd, revised and augmented edition | ||
| 264 | 1 | |a Heidelberg ; Dordrecht ; London ; New York |b Springer |c [2011] | |
| 264 | 4 | |c © 2011 | |
| 300 | |a xxviii, 866 Seiten |b Illustrationen, Diagramme | ||
| 336 | |b txt |2 rdacontent | ||
| 337 | |b n |2 rdamedia | ||
| 338 | |b nc |2 rdacarrier | ||
| 490 | 1 | |a Grundlehren der mathematischen Wissenschaften |v 342 | |
| 500 | |a The 1st edition, published in 1985 in English under Vladimir G. Maz’ja in the Springer Series of Soviet Mathematics was translated from Russian by Tatyana O. Shaposhnikova | ||
| 500 | |a Aus dem Russischen übersetzt | ||
| 650 | 0 | 7 | |a Sobolev-Raum |0 (DE-588)4055345-0 |2 gnd |9 rswk-swf |
| 689 | 0 | 0 | |a Sobolev-Raum |0 (DE-588)4055345-0 |D s |
| 689 | 0 | |5 DE-604 | |
| 776 | 0 | 8 | |i Erscheint auch als |n Online-Ausgabe |z 978-3-642-15564-2 |
| 830 | 0 | |a Grundlehren der mathematischen Wissenschaften |v 342 |w (DE-604)BV000000395 |9 342 | |
| 856 | 4 | 2 | |q text/html |u http://deposit.dnb.de/cgi-bin/dokserv?id=3518834&prov=M&dok_var=1&dok_ext=htm |3 Inhaltstext |
| 856 | 4 | 2 | |m DNB Datenaustausch |q application/pdf |u http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=021173215&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |3 Inhaltsverzeichnis |
| 943 | 1 | |a oai:aleph.bib-bvb.de:BVB01-021173215 | |
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IMAGE 1
CONTENTS
BASIC PROPERTIES OF SOBOLEV SPACES 1
1.1 THE SPACES L L P {Q), V*(UE) AND W L P {Q) 1
1.1.1 NOTATION 1
1.1.2 LOCAL PROPERTIES OF ELEMENTS IN THE SPACE L L P (Q) 2 1.1.3
ABSOLUTE CONTINUITY OF FUNCTIONS IN L P (Q) 4
1.1.4 SPACES W L P {Q) AND V^Q) 7
1.1.5 APPROXIMATION OF FUNCTIONS IN SOBOLEV SPACES BY SMOOTH FUNCTIONS
IN Q 9
1.1.6 APPROXIMATION OF FUNCTIONS IN SOBOLEV SPACES BY FUNCTIONS IN
C(SS) 10
1.1.7 TRANSFORMATION OF COORDINATES IN NORMS OF SOBOLEV SPACES 12
1.1.8 DOMAINS STARSHAPED WITH RESPECT TO A BALL 14 1.1.9 DOMAINS OF THE
CLASS C 0 ' 1 AND DOMAINS HAVING THE CONE PROPERTY . . .' 15
1.1.10 SOBOLEV INTEGRAL REPRESENTATION 16
1.1.11 GENERALIZED POINCARE INEQUALITY 20
1.1.12 COMPLETENESS OF W L P {Q) AND V P L (Q) 22
1.1.13 THE SPACE L L P (N) AND ITS COMPLETENESS 22
1.1.14 ESTIMATE OF INTERMEDIATE DERIVATIVE AND SPACES W L P {Q) AND L L
P {Q) 23
1.1.15 DUALS OF SOBOLEV SPACES 24
1.1.16 EQUIVALENT NORMS-IN W L P {Q) 26
1.1.17 EXTENSION OF FUNCTIONS IN V^Q) ONTO W 1 26
1.1.18 REMOVABLE SETS FOR SOBOLEV FUNCTIONS 28
1.1.19 COMMENTS TO SECT. 1.1 29
1.2 FACTS FROM SET THEORY AND FUNCTION THEORY 32
1.2.1 TWO THEOREMS ON COVERINGS 32
1.2.2 THEOREM ON LEVEL SETS OF A SMOOTH FUNCTION 35
BIBLIOGRAFISCHE INFORMATIONEN HTTP://D-NB.INFO/1005223750
DIGITALISIERT DURCH
IMAGE 2
CONTENTS
1.2.3 REPRESENTATION OF THE LEBESGUE INTEGRAL AS A RIEMANN INTEGRAL
ALONG A HALFAXIS 37
1.2.4 FORMULA FOR THE INTEGRAL OF MODULUS OF THE GRADIENT . . 38 1.2.5
COMMENTS TO SECT. 1.2 39
1.3 SOME INEQUALITIES FOR FUNCTIONS OF ONE VARIABLE 40
1.3.1 BASIC FACTS ON HARDY-TYPE INEQUALITIES 40
1.3.2 TWO-WEIGHT EXTENSIONS OF HARDY'S TYPE INEQUALITY IN THE CASE P Q
42
1.3.3 TWO-WEIGHT EXTENSIONS OF HARDY'S INEQUALITY IN THE CASE P Q 48
1.3.4 HARDY-TYPE INEQUALITIES WITH INDEFINITE WEIGHTS 51 1.3.5 THREE
INEQUALITIES FOR FUNCTIONS ON (0, OO) 57
1.3.6 ESTIMATES FOR DIFFERENTIABLE NONNEGATIVE FUNCTIONS OF ONE VARIABLE
59
1.3.7 COMMENTS TO SECT. 1.3 62
1.4 EMBEDDING THEOREMS OF SOBOLEV TYPE 63
1.4.1 D.R. ADAMS' THEOREM ON RIESZ POTENTIALS 64
1.4.2 ESTIMATE FOR THE NORM IN L Q (M. N , FI) BY THE INTEGRAL OF THE
MODULUS OF THE GRADIENT 67
1.4.3 ESTIMATE FOR THE NORM IN L Q (R N , /X) BY THE INTEGRAL OF THE
MODULUS OF THE ZTH ORDER GRADIENT 70
1.4.4 COROLLARIES OF PREVIOUS RESULTS 72
1.4.5 GENERALIZED SOBOLEV THEOREM 73
1.4.6 COMPACTNESS THEOREMS 76
1.4.7 MULTIPLICATIVE INEQUALITIES 79
1.4.8 COMMENTS TO SECT. 1.4 83
1.5 MORE ON EXTENSION OF FUNCTIONS IN SOBOLEV SPACES 87
1.5.1 SURVEY OF RESULTS AND EXAMPLES OF DOMAINS 87
1.5.2 DOMAINS IN EV P WHICH ARE NOT QUASIDISKS 91
1.5.3 EXTENSION WITH ZERO BOUNDARY DATA 94
1.5.4 COMMENTS TO SECT. 1.5 96
1.6 INEQUALITIES FOR FUNCTIONS WITH ZERO INCOMPLETE CAUCHY DATA . 99
1.6.1 INTEGRAL REPRESENTATION FOR FUNCTIONS OF ONE INDEPENDENT VARIABLE
99
1.6.2 INTEGRAL REPRESENTATION FOR FUNCTIONS OF SEVERAL VARIABLES WITH
ZERO INCOMPLETE CAUCHY DATA 100 1.6.3 EMBEDDING THEOREMS FOR FUNCTIONS
WITH ZERO INCOMPLETE CAUCHY DATA 102
1.6.4 NECESSITY OF THE CONDITION I 2K 105
1.7 DENSITY OF BOUNDED FUNCTIONS IN SOBOLEV SPACES 107 1.7.1 LEMMA ON
APPROXIMATION OF FUNCTIONS IN L P {Q) 107 1.7.2 FUNCTIONS WITH BOUNDED
GRADIENTS ARE NOT ALWAYS DENSE IN L L P (Q) 108
1.7.3 A PLANAR BOUNDED DOMAIN FOR WHICH L\ (J?) N L X (Q) IS NOT DENSE
IN L\(Q) 109
IMAGE 3
CONTENTS XI
1.7.4 DENSITY OF BOUNDED FUNCTIONS IN L P (Q) FOR PARABOLOIDS IN R* 112
1.7.5 COMMENTS TO SECT. 1.7 117
1.8 MAXIMAL ALGEBRA IN W P {Q) 117
1.8.1 MAIN RESULT 117
1.8.2 THE SPACE W%{Q) D L^Q) IS NOT ALWAYS A BANACH ALGEBRA 120
1.8.3 COMMENTS TO SECT. 1.8 121
INEQUALITIES FOR FUNCTIONS VANISHING AT THE BOUNDARY 123 2.1 CONDITIONS
FOR VALIDITY OF INTEGRAL INEQUALITIES (THE CASE P = 1) 124
2.1.1 CRITERION FORMULATED IN TERMS OF ARBITRARY ADMISSIBLE SETS 124
2.1.2 CRITERION FORMULATED IN TERMS OF BALLS FOR UE = R" 126 2.1.3
INEQUALITY INVOLVING THE NORMS IN L Q (F2,/J,) AND L R {Q, V) (CASE P =
1) 127
2.1.4 CASE 6(0,1) 127
2.1.5 INEQUALITY (2.1.10) CONTAINING PARTICULAR MEASURES . . . 132
2.1.6 POWER WEIGHT NORM OF THE GRADIENT ON THE RIGHT-HAND SIDE 133
2.1.7 INEQUALITIES OF HARDY-SOBOLEV TYPE AS COROLLARIES OF THEOREM
2.1.6/1 138
2.1.8 COMMENTS TO SECT. 2.1 140
2.2 (P, #)-CAPACITY 141
2.2.1 DEFINITION AND PROPERTIES OF THE (P, )-CAPACITY 141 2.2.2
EXPRESSION FOR THE (P, $)-CAPACITY CONTAINING AN INTEGRAL OVER LEVEL
SURFACES 144
2.2.3 LOWER ESTIMATES FOR THE (P, )-CAPACITY 146
2.2.4 P-CAPACITY OF A BALL 148
2.2.5 (P, #)-CAPACITY FOR P = 1 149
2.2.6 THE MEASURE M N _I AND 2-CAPACITY 149
2.2.7 COMMENTS TO SECT. 2.2 151
2.3 CONDITIONS FOR VALIDITY OF INTEGRAL INEQUALITIES (THE CASE P 1)
152
2.3.1 THE (P, 2 )-CAPACITARY INEQUALITY 152
2.3.2 CAPACITY MINIMIZING FUNCTION AND ITS APPLICATIONS . . . 156 2.3.3
ESTIMATE FOR A NORM IN A BIRNBAUM-ORLICZ SPACE 157 2.3.4 SOBOLEV TYPE
INEQUALITY AS COROLLARY OF THEOREM 2.3.3 .160 2.3.5 BEST CONSTANT IN THE
SOBOLEV INEQUALITY (P 1) 160 2.3.6 MULTIPLICATIVE INEQUALITY (THE CASE
P 1) 162
2.3.7 ESTIMATE FOR THE NORM IN L Q (Q, /Z) WITH Q P (FIRST NECESSARY
AND SUFFICIENT CONDITION) 165
2.3.8 ESTIMATE FOR THE NORM IN L Q (UE, N) WITH Q P (SECOND NECESSARY
AND SUFFICIENT CONDITION) 167
IMAGE 4
CONTENTS
2.3.9 INEQUALITY WITH THE NORMS IN L Q {Q, FI) AND L R ([2, V) (THE CASE
P 1) 171
2.3.10 ESTIMATE WITH A CHARGE A ON THE LEFT-HAND SIDE 173 2.3.11
MULTIPLICATIVE INEQUALITY WITH THE NORMS IN L Q (Q,II) AND L R (Q, V)
(CASE P 1) 174
2.3.12 ON NASH AND MOSER MULTIPLICATIVE INEQUALITIES 176 2.3.13 COMMENTS
TO SECT. 2.3 177
2.4 CONTINUITY AND COMPACTNESS OF EMBEDDING OPERATORS OF L L P (UE) AND
WJJ(SS) INTO BIRNBAUM-ORLICZ SPACES 179 2.4.1 CONDITIONS FOR BOUNDEDNESS
OF EMBEDDING OPERATORS . 180 2.4.2 CRITERIA FOR COMPACTNESS 182
2.4.3 COMMENTS TO SECT. 2.4 186
2.5 STRUCTURE OF THE NEGATIVE SPECTRUM OF THE MULTIDIMENSIONAL
SCHROEDINGER OPERATOR 188
2.5.1 PRELIMINARIES AND NOTATION 188
2.5.2 POSITIVITY OF THE FORM SI \U, U] 189
2.5.3 SEMIBOUNDEDNESS OF THE SCHROEDINGER OPERATOR 190 2.5.4 DISCRETENESS
OF THE NEGATIVE SPECTRUM 193
2.5.5 DISCRETENESS OF THE NEGATIVE SPECTRUM OF THE OPERATOR S H FOR ALL
H 196
2.5.6 FINITENESS OF THE NEGATIVE SPECTRUM 197
2.5.7 INFINITENESS AND FINITENESS OF THE NEGATIVE SPECTRUM OF THE
OPERATOR S H FOR ALL H 199
2.5.8 PROOFS OF LEMMAS 2.5.1/1 AND 2.5.1/2 200
2.5.9 COMMENTS TO SECT. 2.5 203
2.6 PROPERTIES OF SOBOLEV SPACES GENERATED BY QUADRATIC FORMS WITH
VARIABLE COEFFICIENTS 205
2.6.1 DEGENERATE QUADRATIC FORM 205
2.6.2 COMPLETION IN THE METRIC OF A GENERALIZED DIRICHLET INTEGRAL 208
2.6.3 COMMENTS TO SECT. 2.6 212
2.7 DILATION INVARIANT SHARP HARDY'S INEQUALITIES 213
2.7.1 HARDY'S INEQUALITY WITH SHARP SOBOLEV REMAINDER TERM 213
2.7.2 TWO-WEIGHT HARDY'S INEQUALITIES 214
2.7.3 COMMENTS TO SECT. 2.7 219
2.8 SHARP HARDY-LERAY INEQUALITY FOR AXISYMMETRIC DIVERGENCE-FREE FIELDS
220
2.8.1 STATEMENT OF RESULTS 220
2.8.2 PROOF OF THEOREM 1 222
2.8.3 PROOF OF THEOREM 2 227
2.8.4 COMMENTS TO SECT. 2.8 229
IMAGE 5
CONTENTS XIII
3 CONDUCTOR AND CAPACITARY INEQUALITIES WITH APPLICATIONS TO
SOBOLEV-TYPE EMBEDDINGS 231
3.1 INTRODUCTION 231
3.2 COMPARISON OF INEQUALITIES (3.1.4) AND (3.1.5) 233
3.3 CONDUCTOR INEQUALITY (3.1.1) 234
3.4 APPLICATIONS OF THE CONDUCTOR INEQUALITY (3.1.1) 236 3.5 P-CAPACITY
DEPENDING ON V AND ITS APPLICATIONS TO A CONDUCTOR INEQUALITY AND
INEQUALITY (3.4.1) 241
3.6 COMPACTNESS AND ESSENTIAL NORM 243
3.7 INEQUALITY (3.1.10) WITH INTEGER I 2 245
3.8 TWO-WEIGHT INEQUALITIES INVOLVING FRACTIONAL SOBOLEV NORMS. 249 3.9
COMMENTS TO CHAP. 3 252
4 GENERALIZATIONS FOR FUNCTIONS ON MANIFOLDS AND TOPOLOGICAL SPACES 255
4.1 INTRODUCTION 255
4.2 INTEGRAL INEQUALITIES FOR FUNCTIONS ON RIEMANNIAN MANIFOLDS . 257
4.3 THE FIRST DIRICHLET-LAPLACE EIGENVALUE AND ISOPERIMETRIC CONSTANT
261
4.4 CONDUCTOR INEQUALITIES FOR A DIRICHLET-TYPE INTEGRAL WITH A LOCALITY
PROPERTY 265
4.5 CONDUCTOR INEQUALITY FOR A DIRICHLET-TYPE INTEGRAL WITHOUT LOCALITY
CONDITIONS 270
4.6 SHARP CAPACITARY INEQUALITIES AND THEIR APPLICATIONS 273 4.7
CAPACITARY IMPROVEMENT OF THE FABER-KRAHN INEQUALITY 278 4.8 TWO-WEIGHT
SOBOLEV INEQUALITY WITH SHARP CONSTANT 282 4.9 COMMENTS TO CHAP. 4 286
5 INTEGRABILITY OF FUNCTIONS IN THE SPACE L\(F2) 287
5.1 PRELIMINARIES 288
5.1.1 NOTATION 288
5.1.2 LEMMAS ON APPROXIMATION OF FUNCTIONS IN W PR (Q) AND LL(F2) ' 289
5.2 CLASSES OF SETS ^ A , ^ AND THE EMBEDDING L\{O) C L Q {Q) . 290
5.2.1 CLASSES J A 290
5.2.2 TECHNICAL LEMMA 293
5.2.3 EMBEDDING L\(Q) C L Q (Q) 295
5.2.4 AREA MINIMIZING FUNCTION AM AND EMBEDDING OF L\{Q) INTO L Q ((2)
298
5.2.5 EXAMPLE OF A DOMAIN IN ^\ 299
5.3 SUBAREAL MAPPINGS AND THE CLASSES SS A AND M'A 300
5.3.1 SUBAREAL MAPPINGS 300
5.3.2 ESTIMATE FOR THE FUNCTION A IN TERMS OF SUBAREAL MAPPINGS 302
5.3.3 ESTIMATES FOR THE FUNCTION A FOR SPECIAL DOMAINS 303
IMAGE 6
XIV CONTENTS
5.4 TWO-SIDED ESTIMATES FOR THE FUNCTION A FOR THE DOMAIN IN NIKODYM'S
EXAMPLE 308
5.5 COMPACTNESS OF THE EMBEDDING L\(Q) C L Q {Q) (Q 1) 311 5.5.1 CLASS
J A 311
5.5.2 COMPACTNESS CRITERION 312
5.6 EMBEDDING WL R {Q, DQ) C L Q {Q) 314
5.6.1 CLASS J4,/3 314
5.6.2 EXAMPLES OF SETS IN J(F A SS 315
5.6.3 CONTINUITY OF THE EMBEDDING OPERATOR WL R (Q,DQ) - L Q {Q) 316
5.7 COMMENTS TO CHAP. 5 319
6 INTEGRABILITY OF FUNCTIONS IN THE SPACE L^(FI) 323
6.1 CONDUCTIVITY 324
6.1.1 EQUIVALENCE OF CERTAIN DEFINITIONS OF CONDUCTIVITY 324 6.1.2 SOME
PROPERTIES OF CONDUCTIVITY 326
6.1.3 DIRICHLET PRINCIPLE WITH PRESCRIBED LEVEL SURFACES AND ITS
COROLLARIES 328
6.2 MULTIPLICATIVE INEQUALITY FOR FUNCTIONS WHICH VANISH ON A SUBSET OF
UE 329
6.3 CLASSES OF SETS J V ^ A 331
6.3.1 DEFINITION AND SIMPLE PROPERTIES OF J P , A 331
6.3.2 IDENTITY OF THE CLASSES J^ LJQ AND F A 333
6.3.3 NECESSARY AND SUFFICIENT CONDITION FOR THE VALIDITY OF A
MULTIPLICATIVE INEQUALITY FOR FUNCTIONS IN W P S (Q) . 334 6.3.4
CRITERION FOR THE EMBEDDING W^ S (Q) C L Q , (Q), P Q* ' 336
6.3.5 FUNCTION VM, P AND THE RELATIONSHIP OF THE CLASSES JV AND J A 337
6.3.6 ESTIMATES FOR THE CONDUCTIVITY MINIMIZING FUNCTION VM,? FOR
CERTAIN DOMAINS 338
6.4 EMBEDDING H^ S (J?) C L Q * (UE) FOR Q* P 341
6.4.1 ESTIMATE FOR THE NORM IN L Q *(F2) WITH Q* P FOR FUNCTIONS WHICH
VANISH ON A SUBSET OF Q 341
6.4.2 CLASS J^, Q AND THE EMBEDDING WL S {Q) C L Q . (UE) FOR 0 Q* P
' 342
6.4.3 EMBEDDING L),(Q) C L Q . (UE) FOR A DOMAIN WITH FINITE VOLUME 343
6.4.4 SUFFICIENT CONDITION FOR BELONGING TO J$? PTA 345
6.4.5 NECESSARY CONDITIONS FOR BELONGING TO THE CLASSES J^ PT A AND J F
PTA 345
6.4.6 EXAMPLES OF DOMAINS IN J^, )Q 347
6.4.7 OTHER DESCRIPTIONS OF THE CLASSES J P , A AND J ^ IQ 348
6.4.8 INTEGRAL INEQUALITIES FOR DOMAINS WITH POWER C U S P S . . . 350
IMAGE 7
CONTENTS XV
6.5 MORE ON THE NIKODYM EXAMPLE 352
6.6 SOME GENERALIZATIONS 360
6.7 INCLUSION W PR (Q) C L Q {Q) (R Q) FOR DOMAINS WITH INFINITE
VOLUME 364
OO OO
6.7.1 CLASSES J A AND J^, Q 364
6.7.2 EMBEDDING W^ R (Q) C L Q (Q) (R Q) 367
OO
6.7.3 EXAMPLE OF A DOMAIN IN THE CLASS Y P A 368
(O)
6.7.4 SPACE L L P (/?) AND ITS EMBEDDING INTO L Q (Q) 370
6.7.5 POINCARE-TYPE INEQUALITY FOR DOMAINS WITH INFINITE VOLUME 371
6.8 COMPACTNESS OF THE EMBEDDING L P {Q) C L Q (Q) 374
6.8.1 CLASS JV 374
6.8.2 COMPACTNESS CRITERIA 375
6.8.3 SUFFICIENT CONDITIONS FOR COMPACTNESS OF THE EMBEDDING L L P (Q) C
I , . ( SS) 376
6.8.4 COMPACTNESS THEOREM FOR AN ARBITRARY DOMAIN WITH FINITE VOLUME 377
6.8.5 EXAMPLES OF DOMAINS IN THE CLASS J^ PJQ 378
6.9 EMBEDDING L L P (Q) C L Q {Q) 379
6.10 APPLICATIONS TO THE NEUMANN PROBLEM FOR STRONGLY ELLIPTIC OPERATORS
380
6.10.1 SECOND-ORDER OPERATORS 381
6.10.2 NEUMANN PROBLEM FOR OPERATORS OF ARBITRARY ORDER . . 382 6.10.3
NEUMANN PROBLEM FOR A SPECIAL DOMAIN 385 6.10.4 COUNTEREXAMPLE TO
INEQUALITY (6.10.7) 389
6.11 INEQUALITIES CONTAINING INTEGRALS OVER THE BOUNDARY 390 6.11.1
EMBEDDING W^Q, DQ) C L G (J?) 390
6.11.2 CLASSES J^.T 15 AND J^~ X) 393
6.11.3 EXAMPLES OF DOMAINS IN J ^ T^ AND FI N ~ L) 394
6.11.4 ESTIMATES FOR THE NORM IN L Q (DF2) 395
6.11.5 CLASS J? P *OC AND COMPACTNESS THEOREMS 397 6.11.6 CRITERIA OF
SOLVABILITY OF BOUNDARY VALUE PROBLEMS FOR SECOND-ORDER ELLIPTIC
EQUATIONS 399
6.12 COMMENTS TO CHAP. 6 401
CONTINUITY AND BOUNDEDNESS OF FUNCTIONS IN SOBOLEV SPACES 405
7.1 THE EMBEDDING W^UE) C C(Q) N (#) 406
7.1.1 CRITERIA FOR CONTINUITY OF EMBEDDING OPERATORS OF W P L {Q) AND
L\(!2) INTO C{Q) N L X {Q) 406
7.1.2 SUFFICIENT CONDITION IN TERMS OF THE ISOPERIMETRIC FUNCTION FOR
THE EMBEDDING W P L {Q) C C{Q) N L^Q) . 409
IMAGE 8
CONTENTS
7.1.3 ISOPERIMETRIC FUNCTION AND A BREZIS-GALLOUET- WAINGER-TYPE
INEQUALITY 410
7.2 MULTIPLICATIVE ESTIMATE FOR MODULUS OF A FUNCTION IN W P (F2) .412
7.2.1 CONDITIONS FOR VALIDITY OF A MULTIPLICATIVE INEQUALITY. 412
7.2.2 MULTIPLICATIVE INEQUALITY IN THE LIMIT CASE R = (P - N)/N 414
7.3 CONTINUITY MODULUS OF FUNCTIONS IN L P {Q) 416
7.4 BOUNDEDNESS OF FUNCTIONS WITH DERIVATIVES IN BIRNBAUM- ORLICZ SPACES
419
7.5 COMPACTNESS OF THE EMBEDDING W^{UE) C C(F2) F) L^Q) 422 7.5.1
COMPACTNESS CRITERION 422
7.5.2 SUFFICIENT CONDITION FOR THE COMPACTNESS IN TERMS OF THE
ISOPERIMETRIC FUNCTION 423
7.5.3 DOMAIN FOR WHICH THE EMBEDDING OPERATOR OF W P (Q) INTO C(2) N L
OO (Q) IS BOUNDED BUT NOT COMPACT 424 7.6 GENERALIZATIONS TO SOBOLEV
SPACES OF AN ARBITRARY INTEGER ORDER 426
7.6.1 THE (P, ^-CONDUCTIVITY 426
7.6.2 EMBEDDING L L P (UE) C C{Q) N L^O) 427
7.6.3 EMBEDDING V P \Q) C C(Q) D L^TI) 428
7.6.4 COMPACTNESS OF THE EMBEDDING L L P {Q) C C{Q) N LOO(SS) 429
7.6.5 SUFFICIENT CONDITIONS FOR THE CONTINUITY AND THE COMPACTNESS OF
THE EMBEDDING L L P (O) C C{Q) N LOO(FL) 430
7.6.6 EMBEDDING OPERATORS FOR THE SPACE W L P {Q) N W^(UE), L 2K 432
7.7 COMMENTS TO CHAP. 7 434
LOCALIZATION MODULI OF SOBOLEV EMBEDDINGS FOR GENERAL DOMAINS 435
8.1 LOCALIZATION MODULI AND THEIR PROPERTIES 437
8.2 COUNTEREXAMPLE FOR THE CASE P = Q 442
8.3 CRITICAL SOBOLEV EXPONENT 444
8.4 GENERALIZATION 446
8.5 MEASURES OF NONCOMPACTNESS FOR POWER CUSP-SHAPED DOMAINS 447
8.6 FINITENESS OF THE NEGATIVE SPECTRUM OF A SCHROEDINGER OPERATOR ON
/3-CUSP DOMAINS 452
8.7 RELATIONS OF MEASURES OF NONCOMPACTNESS WITH LOCAL ISOCONDUCTIVITY
AND ISOPERIMETRIC CONSTANTS 456
8.8 COMMENTS TO CHAP. 8 457
IMAGE 9
CONTENTS XVII
SPACE OF FUNCTIONS OF BOUNDED VARIATION 459
9.1 PROPERTIES OF THE SET PERIMETER AND FUNCTIONS IN BV(Q) 459 9.1.1
DEFINITIONS OF THE SPACE BV(FI) AND OF THE RELATIVE PERIMETER 459
9.1.2 APPROXIMATION OF FUNCTIONS IN BV(Q) 460
9.1.3 APPROXIMATION OF SETS WITH FINITE PERIMETER 463 9.1.4 COMPACTNESS
OF THE FAMILY OF SETS WITH UNIFORMLY BOUNDED RELATIVE PERIMETERS 464
9.1.5 ISOPERIMETRIC INEQUALITY 464
9.1.6 INTEGRAL FORMULA FOR THE NORM IN BV(S1) 465
9.1.7 EMBEDDING BV{Q) C L Q (Q) 466
9.2 GAUSS-GREEN FORMULA FOR LIPSCHITZ FUNCTIONS 467
9.2.1 NORMAL IN THE SENSE OF FEDERER AND REDUCED BOUNDARY 467
9.2.2 GAUSS-GREEN FORMULA 467
9.2.3 SEVERAL AUXILIARY ASSERTIONS 468
9.2.4 STUDY OF THE SET N 470
9.2.5 RELATIONS BETWEEN VARVX ? AND S ON DS 473
9.3 EXTENSION OF FUNCTIONS IN EV{Q) ONTO R N 477
9.3.1 PROOF OF NECESSITY OF (9.3.2) 478
9.3.2 THREE LEMMAS ON PCN[SS) 478
9.3.3 PROOF OF SUFFICIENCY OF (9.3.2) 480
9.3.4 EQUIVALENT STATEMENT OF THEOREM 9.3 482
9.3.5 ONE MORE EXTENSION THEOREM 483
9.4 EXACT CONSTANTS FOR CERTAIN CONVEX DOMAINS 484
9.4.1 LEMMAS ON APPROXIMATIONS BY POLYHEDRA 484 9.4.2 PROPERTY OF P C Q
486
9.4.3 EXPRESSION FOR THE SET FUNCTION TQ{) FOR A CONVEX DOMAIN 486
9.4.4 THE FUNCTION \FI\ FOR A CONVEX DOMAIN 487
9.5 ROUGH TRACE OF FUNCTIONS IN BV{Q) AND CERTAIN INTEGRAL INEQUALITIES
489
9.5.1 DEFINITION OF THE ROUGH TRACE AND ITS PROPERTIES 489 9.5.2
INTEGRABILITY OF THE ROUGH TRACE 492
9.5.3 EXACT CONSTANTS IN CERTAIN INTEGRAL ESTIMATES FOR THE ROUGH TRACE
493
9.5.4 MORE ON INTEGRABILITY OF THE ROUGH TRACE 495
9.5.5 EXTENSION OF A FUNCTION IN BV{Q) TO CQ BY A CONSTANT 496
9.5.6 MULTIPLICATIVE ESTIMATES FOR THE ROUGH TRACE 497 9.5.7 ESTIMATE
FOR THE NORM IN L*/( N _ 1 )(J?) OF A FUNCTION IN BV(Q) WITH INTEGRABLE
ROUGH TRACE 499
9.6 TRACES OF FUNCTIONS IN BV{Q) ON THE BOUNDARY AND GAUSS-GREEN FORMULA
500
9.6.1 DEFINITION OF THE TRACE 500
IMAGE 10
XVIII CONTENTS
9.6.2 COINCIDENCE OF THE TRACE AND THE ROUGH TRACE 501 9.6.3 TRACE OF
THE CHARACTERISTIC FUNCTION 504
9.6.4 INTEGRABILITY OF THE TRACE OF A FUNCTION IN BV(Q) 504 9.6.5
GAUSS-GREEN FORMULA FOR FUNCTIONS IN BV(FI) 505 9.7 COMMENTS TO CHAP. 9
507
10 CERTAIN FUNCTION SPACES, CAPACITIES, AND POTENTIALS 511 10.1 SPACES
OF FUNCTIONS DIFFERENTIABLE OF ARBITRARY POSITIVE ORDER . 512 10.1.1
SPACES W' P , W L P , B L P , B L P FOR I 0 512
10.1.2 RIESZ AND BESSEL POTENTIAL SPACES 516
10.1.3 OTHER PROPERTIES OF THE INTRODUCED FUNCTION SPACES . . 519 10.2
BOURGAIN, BREZIS, AND MIRONESCU THEOREM CONCERNING LIMITING EMBEDDINGS
OF FRACTIONAL SOBOLEV SPACES 521 10.2.1 INTRODUCTION 521
10.2.2 HARDY-TYPE INEQUALITIES 522
10.2.3 SOBOLEV EMBEDDINGS 528
10.2.4 ASYMPTOTICS OF THE NORM IN V*(K N ) AS S J 0 528 10.3 ON THE
BREZIS AND MIRONESCU CONJECTURE CONCERNING A GAGLIARDO-NIRENBERG
INEQUALITY FOR FRACTIONAL SOBOLEV NORMS . 530 10.3.1 INTRODUCTION 530
10.3.2 MAIN THEOREM 531
10.4 SOME FACTS FROM NONLINEAR POTENTIAL THEORY 536
10.4.1 CAPACITY CAP(E, S L P ) AND ITS PROPERTIES 536
10.4.2 NONLINEAR POTENTIALS 538
10.4.3 METRIC PROPERTIES OF CAPACITY 541
10.4.4 REFINED FUNCTIONS 544
10.5 COMMENTS TO CHAP. 10 545
11 CAPACITARY AND TRACE INEQUALITIES FOR FUNCTIONS IN R N WITH
DERIVATIVES OF AN ARBITRARY ORDER 549
11.1 DESCRIPTION OF RESULTS 549
11.2 CAPACITARY INEQUALITY OF AN ARBITRARY ORDER 552
11.2.1 A PROOF BASED ON THE SMOOTH TRUNCATION OF A POTENTIAL 552
11.2.2 A PROOF BASED ON THE MAXIMUM PRINCIPLE FOR NONLINEAR POTENTIALS
554
11.3 CONDITIONS FOR THE VALIDITY OF EMBEDDING THEOREMS IN TERMS OF
ISOCAPACITARY INEQUALITIES 556
11.4 COUNTEREXAMPLE TO THE CAPACITARY INEQUALITY FOR THE NORM IN L\[SS)
558
11.5 BALL AND POINTWISE CRITERIA 564
11.6 CONDITIONS FOR EMBEDDING INTO L Q (FI) FOR P Q 0 570 11.6.1
CRITERION IN TERMS OF THE CAPACITY MINIMIZING FUNCTION 570
11.6.2 TWO SIMPLE CASES 574
IMAGE 11
CONTENTS XIX
11.7 CARTAN-TYPE THEOREM AND ESTIMATES FOR CAPACITIES 575 11.8 EMBEDDING
THEOREMS FOR THE SPACE S L P (CONDITIONS IN TERMS OF BALLS, P 1) 579
11.9 APPLICATIONS 582
11.9.1 COMPACTNESS CRITERIA 582
11.9.2 EQUIVALENCE OF CONTINUITY AND COMPACTNESS OF THE EMBEDDING H L P
C L G (SS) FOR P Q 583
11.9.3 APPLICATIONS TO THE THEORY OF ELLIPTIC OPERATORS . . . 586
11.9.4 CRITERIA FOR THE RELLICH-KATO INEQUALITY 586 11.10 EMBEDDING
THEOREMS FOR P = 1 588
11.10.1 INTEGRABILITY WITH RESPECT TO A MEASURE 588 11.10.2 CRITERION
FOR AN UPPER ESTIMATE OF A DIFFERENCE SEMINORM (THE CASE P = 1) 590
11.10.3 EMBEDDING INTO A RIESZ POTENTIAL SPACE 596 11.11 CRITERIA FOR AN
UPPER ESTIMATE OF A DIFFERENCE SEMINORM (THE CASE P 1) , 597
11.11.1 CASE Q P 597
11.11.2 CAPACITARY SUFFICIENT CONDITION IN THE CASE Q = P . . 603 11.12
COMMENTS TO CHAP. 11 607
12 POINTWISE INTERPOLATION INEQUALITIES FOR DERIVATIVES AND POTENTIALS
611
12.1 POINTWISE INTERPOLATION INEQUALITIES FOR RIESZ AND BESSEL
POTENTIALS 612
12.1.1 ESTIMATE FOR THE MAXIMAL OPERATOR OF A CONVOLUTION 612 12.1.2
POINTWISE INTERPOLATION INEQUALITY FOR RIESZ POTENTIALS 613
12.1.3 ESTIMATES FOR \J- WXP\ 6 14
12.1.4 ESTIMATES FOR \J- W (5 - X P )\ 619
12.1.5 POINTWISE INTERPOLATION INEQUALITY FOR BESSEL POTENTIALS 620
12.1.6 POINTWISE ESTIMATES INVOLVING MVKU AND A L U 622 12.1.7
APPLICATION: WEIGHTED NORM INTERPOLATION INEQUALITIES FOR POTENTIALS 623
12.2 SHARP POINTWISE INEQUALITIES FOR VU 624
12.2.1 THE CASE OF NONNEGATIVE FUNCTIONS 624
12.2.2 FUNCTIONS WITH UNRESTRICTED SIGN. MAIN RESULT 624 12.2.3 PROOF OF
INEQUALITY (12.2.6) 626
12.2.4 PROOF OF SHARPNESS 627
12.2.5 PARTICULAR CASE U(R) = R A ,A 0 634
12.2.6 ONE-DIMENSIONAL CASE 636
12.3 POINTWISE INTERPOLATION INEQUALITIES INVOLVING "FRACTIONAL
DERIVATIVES" 638
12.3.1 INEQUALITIES WITH FRACTIONAL DERIVATIVES ON THE RIGHT-HAND SIDES
638
IMAGE 12
XX CONTENTS
12.3.2 INEQUALITY WITH A FRACTIONAL DERIVATIVE OPERATOR ON THE LEFT-HAND
SIDE 641
12.3.3 APPLICATION: WEIGHTED GAGLIARDO-NIRENBERG-TYPE INEQUALITIES FOR
DERIVATIVES 643
12.4 APPLICATION OF (12.3.11) TO COMPOSITION OPERATOR IN FRACTIONAL
SOBOLEV SPACES 643
12.4.1 INTRODUCTION 643
12.4.2 PROOF OF INEQUALITY (12.4.1) 645
12.4.3 CONTINUITY OF THE MAP (12.4.2) 648
12.5 COMMENTS TO CHAP. 12 653
13 A VARIANT OF CAPACITY 657
13.1 CAPACITY CAP 657
13.1.1 SIMPLE PROPERTIES OF CAP(E, L L P (Q)) 657
13.1.2 CAPACITY OF A CONTINUUM 660
13.1.3 CAPACITY OF A BOUNDED CYLINDER 662
13.1.4 SETS OF ZERO CAPACITY CAP(-, W P ) 663
13.2 ON (P, /)-POLAR SETS 663
13.3 EQUIVALENCE OF TWO CAPACITIES 664
13.4 REMOVABLE SINGULARITIES OF Z-HARMONIC FUNCTIONS IN L* 666 13.5
COMMENTS TO CHAP. 13 668
14 INTEGRAL INEQUALITY FOR FUNCTIONS ON A CUBE 669
14.1 CONNECTION BETWEEN THE BEST CONSTANT AND CAPACITY (CASE K = 1) 670
14.1.1 DEFINITION OF A (P, ^-NEGLIGIBLE SET 670
14.1.2 MAIN THEOREM 670
14.1.3 VARIANT OF THEOREM 14.1.2 AND ITS COROLLARIES 673 14.2 CONNECTION
BETWEEN BEST CONSTANT AND THE (P, Z)-INNER DIAMETER (CASE K = 1) 675
14.2.1 SET FUNCTION A^ G (G) 675
14.2.2 DEFINITION OF THE (P, Z)-INNER DIAMETER 676
14.2.3 ESTIMATES FOR THE BEST CONSTANT IN (14.1.3) BY THE (P, Z)-INNER
DIAMETER 676
14.3 ESTIMATES FOR THE BEST CONSTANT C IN THE GENERAL CASE 679 14.3.1
NECESSARY AND SUFFICIENT CONDITION FOR VALIDITY OF THE BASIC INEQUALITY
679
14.3.2 POLYNOMIAL CAPACITIES OF FUNCTION CLASSES 680 14.3.3 ESTIMATES
FOR THE BEST CONSTANT C IN THE BASIC INEQUALITY 681
14.3.4 CLASS C 0 (E) AND CAPACITY CAP FC (E, L P (Q 2D )) 684 14.3.5
LOWER BOUND FOR CAP FC 685
14.3.6 ESTIMATES FOR THE BEST CONSTANT IN THE CASE OF SMALL (P, Z)-INNER
DIAMETER 687
IMAGE 13
CONTENTS XXI
14.3.7 A LOGARITHMIC SOBOLEV INEQUALITY WITH APPLICATION TO THE
UNIQUENESS THEOREM FOR ANALYTIC FUNCTIONS IN THE CLASS L P (U) 689
14.4 COMMENTS TO CHAP. 14 691
15 EMBEDDING OF THE SPACE L L P (F2) INTO OTHER FUNCTION SPACES 693
15.1 PRELIMINARIES 693
15.2 EMBEDDING L L P {Q) C &{Q) 694
15.2.1 AUXILIARY ASSERTIONS 694
15.2.2 CASE Q = W 1 696
15.2.3 CASE N = PI, P 1 697
15.2.4 CASE N PL AND NONINTEGER N/P 697
15.2.5 CASE N PI, 1 P OO, AND INTEGER N/P 698
15.3 EMBEDDING L L P {Q) C L Q {N,\OC) 701
15.4 EMBEDDING L L P {Q) C L Q {Q) (THE CASE P Q) 703
15.4.1 A CONDITION IN TERMS OF THE (P, Z)-INNER DIAMETER . . 703 15.4.2
A CONDITION IN TERMS OF CAPACITY 704
15.5 EMBEDDING L L P (Q) C L Q (UE) (THE CASE P Q 1) 707
15.5.1 DEFINITIONS AND LEMMAS 707
15.5.2 BASIC RESULT 710
15.5.3 EMBEDDING L L P {Q) C L Q (Q) FOR AN "INFINITE FUNNEL" . 712 15.6
COMPACTNESS OF THE EMBEDDING L L P (Q) C L Q {Q) 714 15.6.1 CASE P Q
714
15.6.2 CASE P Q 715
15.7 APPLICATION TO THE DIRICHLET PROBLEM FOR A STRONGLY ELLIPTIC
OPERATOR 716
15.7.1 DIRICHLET PROBLEM WITH NONHOMOGENEOUS BOUNDARY DATA 717
15.7.2 DIRICHLET PROBLEM WITH HOMOGENEOUS BOUNDARY DATA 718
15.7.3 DISCRETENESS OF THE SPECTRUM OF THE DIRICHLET PROBLEM 719
15.7.4 DIRICHLET PROBLEM FOR A NONSELFADJOINT OPERATOR 719 15.8
APPLICATIONS TO THE THEORY OF QUASILINEAR ELLIPTIC EQUATIONS . 721
15.8.1 SOLVABILITY OF THE DIRICHLET PROBLEM FOR QUASILINEAR EQUATIONS IN
UNBOUNDED DOMAINS 721
15.8.2 A WEIGHTED MULTIPLICATIVE INEQUALITY 725
15.8.3 UNIQUENESS OF A SOLUTION TO THE DIRICHLET PROBLEM WITH AN
EXCEPTIONAL SET FOR EQUATIONS OF ARBITRARY ORDER 727
15.8.4 UNIQUENESS OF A SOLUTION TO THE NEUMANN PROBLEM FOR QUASILINEAR
SECOND-ORDER EQUATION 730 15.9 COMMENTS TO CHAP. 15 733
IMAGE 14
XXII CONTENTS
16 EMBEDDING L L P {FL, U) C W*(F2) 737
16.1 AUXILIARY ASSERTIONS 737
16.2 CONTINUITY OF THE EMBEDDING OPERATOR L L P (Q, V) -* W*{Q) 739
16.3 COMPACTNESS OF THE EMBEDDING OPERATOR L L P (Q, V) -* W*(Q) 742
16.3.1 ESSENTIAL NORM OF THE EMBEDDING OPERATOR 742 16.3.2 CRITERIA FOR
COMPACTNESS 744
16.4 CLOSABILITY OF EMBEDDING OPERATORS 746
16.5 APPLICATION: POSITIVE DEFINITENESS AND DISCRETENESS OF THE SPECTRUM
OF A STRONGLY ELLIPTIC OPERATOR 749
16.6 COMMENTS TO CHAP. 16 751
17 APPROXIMATION IN WEIGHTED SOBOLEV SPACES 755
17.1 MAIN RESULTS AND APPLICATIONS 755
17.2 CAPACITIES 757
17.3 APPLICATIONS OF LEMMA 17.2/3 761
17.4 PROOF OF THEOREM 17.1 765
17.5 COMMENTS TO CHAP. 17 768
18 SPECTRUM OF THE SCHROEDINGER OPERATOR AND THE DIRICHLET LAPLACIAN 769
18.1 MAIN RESULTS ON THE SCHROEDINGER OPERATOR 770
18.2 DISCRETENESS OF SPECTRUM: NECESSITY 773
18.3 DISCRETENESS OF SPECTRUM: SUFFICIENCY 781
18.4 A SUFFICIENCY EXAMPLE 783
18.5 POSITIVITY OF H W 787
18.6 STRUCTURE OF THE ESSENTIAL SPECTRUM OF HY 787
18.7 TWO-SIDED ESTIMATES OF THE FIRST EIGENVALUE OF THE DIRICHLET
LAPLACIAN 789
18.7.1 MAIN RESULT 789
18.7.2 LOWER BOUND 790
18.7.3 UPPER BOUND 795
18.7.4 COMMENTS TO CHAP. 18 800
REFERENCES 803
LIST OF SYMBOLS 849
SUBJECT INDEX 853
AUTHOR INDEX 859 |
| any_adam_object | 1 |
| author | Mazʹja, Vladimir Gilelevič 1937- |
| author_GND | (DE-588)121490602 |
| author_facet | Mazʹja, Vladimir Gilelevič 1937- |
| author_role | aut |
| author_sort | Mazʹja, Vladimir Gilelevič 1937- |
| author_variant | v g m vg vgm |
| building | Verbundindex |
| bvnumber | BV037260069 |
| classification_rvk | SK 600 |
| ctrlnum | (OCoLC)706790293 (DE-599)DNB1005223750 |
| dewey-full | 515.782 |
| dewey-hundreds | 500 - Natural sciences and mathematics |
| dewey-ones | 515 - Analysis |
| dewey-raw | 515.782 |
| dewey-search | 515.782 |
| dewey-sort | 3515.782 |
| dewey-tens | 510 - Mathematics |
| discipline | Mathematik |
| edition | 2nd, revised and augmented edition |
| format | Book |
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| id | DE-604.BV037260069 |
| illustrated | Illustrated |
| indexdate | 2024-07-20T11:00:32Z |
| institution | BVB |
| isbn | 9783642155635 9783662517291 |
| language | English Russian |
| oai_aleph_id | oai:aleph.bib-bvb.de:BVB01-021173215 |
| oclc_num | 706790293 |
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| owner_facet | DE-384 DE-20 DE-83 DE-634 DE-19 DE-BY-UBM DE-824 DE-188 DE-11 DE-29T |
| physical | xxviii, 866 Seiten Illustrationen, Diagramme |
| publishDate | 2011 |
| publishDateSearch | 2011 |
| publishDateSort | 2011 |
| publisher | Springer |
| record_format | marc |
| series | Grundlehren der mathematischen Wissenschaften |
| series2 | Grundlehren der mathematischen Wissenschaften |
| spelling | Mazʹja, Vladimir Gilelevič 1937- Verfasser (DE-588)121490602 aut Sobolev spaces with applications to elliptic partial differential equations Vladimir Maz'ya 2nd, revised and augmented edition Heidelberg ; Dordrecht ; London ; New York Springer [2011] © 2011 xxviii, 866 Seiten Illustrationen, Diagramme txt rdacontent n rdamedia nc rdacarrier Grundlehren der mathematischen Wissenschaften 342 The 1st edition, published in 1985 in English under Vladimir G. Maz’ja in the Springer Series of Soviet Mathematics was translated from Russian by Tatyana O. Shaposhnikova Aus dem Russischen übersetzt Sobolev-Raum (DE-588)4055345-0 gnd rswk-swf Sobolev-Raum (DE-588)4055345-0 s DE-604 Erscheint auch als Online-Ausgabe 978-3-642-15564-2 Grundlehren der mathematischen Wissenschaften 342 (DE-604)BV000000395 342 text/html http://deposit.dnb.de/cgi-bin/dokserv?id=3518834&prov=M&dok_var=1&dok_ext=htm Inhaltstext DNB Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=021173215&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Mazʹja, Vladimir Gilelevič 1937- Sobolev spaces with applications to elliptic partial differential equations Grundlehren der mathematischen Wissenschaften Sobolev-Raum (DE-588)4055345-0 gnd |
| subject_GND | (DE-588)4055345-0 |
| title | Sobolev spaces with applications to elliptic partial differential equations |
| title_auth | Sobolev spaces with applications to elliptic partial differential equations |
| title_exact_search | Sobolev spaces with applications to elliptic partial differential equations |
| title_full | Sobolev spaces with applications to elliptic partial differential equations Vladimir Maz'ya |
| title_fullStr | Sobolev spaces with applications to elliptic partial differential equations Vladimir Maz'ya |
| title_full_unstemmed | Sobolev spaces with applications to elliptic partial differential equations Vladimir Maz'ya |
| title_short | Sobolev spaces |
| title_sort | sobolev spaces with applications to elliptic partial differential equations |
| title_sub | with applications to elliptic partial differential equations |
| topic | Sobolev-Raum (DE-588)4055345-0 gnd |
| topic_facet | Sobolev-Raum |
| url | http://deposit.dnb.de/cgi-bin/dokserv?id=3518834&prov=M&dok_var=1&dok_ext=htm http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=021173215&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
| volume_link | (DE-604)BV000000395 |
| work_keys_str_mv | AT mazʹjavladimirgilelevic sobolevspaceswithapplicationstoellipticpartialdifferentialequations |