Differentiable functions on bad domains:
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| Main Authors: | , |
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| Format: | Book |
| Language: | English |
| Published: |
Singapore [u.a.]
World Scientific
1997
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| Subjects: | |
| Online Access: | Inhaltsverzeichnis |
| Physical Description: | XVIII, 481 S. graph. Darst. |
| ISBN: | 9810227671 |
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MARC
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| 245 | 1 | 0 | |a Differentiable functions on bad domains |c Vladimir G. Maz'ya ; Sergei V. Poborchi |
| 264 | 1 | |a Singapore [u.a.] |b World Scientific |c 1997 | |
| 300 | |a XVIII, 481 S. |b graph. Darst. | ||
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| adam_text | DIFFERENTIABLE FUNCTIONS ON BAD DOMAINS VLADIMIR G MAZ YA LINKOPING
UNIVERSITY, SWEDEN SERGEI V POBORCHI ST PETERSBURG STATE UNIVERSITY,
RUSSIA WORLD SCIENTIFIC SINGAPORE * NEW JERSEY * LONDON * HONG KONG
TABLE OF CONTENTS I. INTRODUCTION TO SOBOLEV SPACES FOR DOMAINS 1. BASIC
PROPERTIES OF SOBOLEV SPACES 1 1.1. PRELIMINARIES 1 1.1.1. NOTATION 1
1.1.2. THE SPACE L P AND INTEGRAL INEQUALITIES 3 1.2. FUNCTIONS WITH
GENERALIZED DERIVATIVES 7 1.2.1. MOLLIFICATION 7 1.2.2. GENERALIZED
DERIVATIVES 9 1.2.3. THE SPACES L LP (CL), W P (N), V PL (CI) 12 1.2.4.
ABSOLUTE CONTINUITY OF FUNCTIONS IN L P (Q) 14 1.2.5. ON REMOVABLE
SINGULARITIES FOR FUNCTIONS IN V PL (Q.) 16 1.3. CLASSES OF DOMAINS 17
1.3.1. DOMAINS OF CLASS C AND DOMAINS HAVING THE SEGMENT PROPERTY 17
1.3.2. DOMAINS STARSHAPED WITH RESPECT TO A BALL AND DOMAINS OF CLASS C
0 1 X 20 1.3.3. DOMAINS HAVING THE CONE PROPERTY 22 1.3.4. DOMAINS
OF CLASS C 0 1 AND LIPSCHITZ DOMAINS 23 1.4. DENSITY OF SMOOTH
FUNCTIONS IN SOBOLEV SPACES 24 1.4.1. APPROXIMATION OF FUNCTIONS IN
SOBOLEV SPACES BY FUNCTIONS IN C(FI) 24 1.4.2. APPROXIMATION BY
FUNCTIONS IN C(Q.) 27 1.4.3. DENSITY OF BOUNDED SMOOTH FUNCTIONS IN L
P (CL) AND W^{0) 29 1.5. POINCARE S INEQUALITY AND EQUIVALENT NORMS IN
SOBOLEV SPACES 31 XII TABLE OF CONTENTS 1.5.1. SOBOLEV S INTEGRAL
REPRESENTATION 31 1.5.2. GENERALIZED POINCARE INEQUALITY 35 1.5.3. THE
SPACE I/J,(FT) AND NORMINGS IN L P (Q) 39 1.5.4. EQUIVALENT NORMS IN W P
(CL) , 40 1.6. EXTENDABILITY OF FUNCTIONS IN SOBOLEV SPACES 42 1.6.1.
EXTENSION ACROSS THE PLANE PART OF A BOUNDARY 42 1.6.2. DOMAINS OF
CLASS EV^ 45 1.7. CHANGE OF COORDINATES FOR SOBOLEV FUNCTIONS 46 1.8.
SUMMABILITY AND CONTINUITY OF FUNCTIONS IN SOBOLEV SPACES 47 1.8.1. ON
CONTINUITY OF THE IMBEDDING OPERATOR: WJ,(R N ) - L Q (R N ) 47 1.8.2.
SOBOLEV S THEOREM 54 1.9. EQUIVALENCE OF INTEGRAL AND ISOPERIMETRIC
INEQUALITIES 57 1.10. COMPACTNESS THEOREMS 60 1.11. THE MAXIMAL ALGEBRA
IN W^(FT) 64 1.12. APPLICATION TO THE NEUMANN PROBLEM FOR ELLIPTIC
OPERATORS OF ARBITRARY ORDER 68 1.12.1. NECESSARY AND SUFFICIENT
CONDITION FOR THE CONTINUITY OF THE IMBEDDING L LP {Q) C L Q (Q.) 68
1.12.2. SOLVABILITY OF THE NEUMANN PROBLEM 69 EXERCISES FOR CHAPTER 1 .^
73 COMMENTS TO CHAPTER 1 78 2. EXAMPLES OF BAD DOMAINS IN THE THEORY
OF SOBOLEV SPACES 89 2.1. THE PROPERTY 9FT * DCL DOES NOT ENSURE THE
DENSITY OF C(FT) IN SOBOLEV SPACES 89 2.2. FUNCTIONS WITH BOUNDED
GRADIENTS ARE NOT ALWAYS DENSE IN I(Q) 92 2.3. A PLANAR BOUNDED DOMAIN
FOR WHICH IF (FT) N L^CL) IS NOT DENSE IN L (FT) 94 TABLE OF CONTENTS
XIII 2.4. ON DENSITY OF BOUNDED FUNCTIONS IN L^(FT) FOR PARABOLOIDS IN
R 98 2.5. IMBEDDING AND COMPACTNESS PROPERTIES MAY- FAIL FOR THE
INTERSECTION OF GOOD DOMAINS 103 2.6. A DOMAIN FOR WHICH THE IMBEDDING
W PX (FT) C C(FT) N LOO (FT) IS CONTINUOUS BUT NONCOMPACT 105 2.7.
NIKODYM S DOMAIN 107 2.7.1. A DOMAIN WITH THE PROPERTY L{,(FT) L,(FT)
FOR I = 1,2,..., Q 0 AND P 6 [1, OO) 107 2.7.2. A DOMAIN FOR WHICH V
PL (Q.) IS NONCOMPACTLY IMBEDDED INTO L P (FT) FOR P * [1, OO] AND I =
1,2, 108 2.7.3. EQUIVALENCE OF THE IMBEDDINGS L P (FT) C L G (FL) AND
L*(FT) C L G (FT) 109 2.7.4. THE NEUMANN PROBLEM FOR NIKODYM S DOMAIN
114 2.8. THE SPACE W%{1) N LOO (FT) IS NOT ALWAYS A BANACH ALGEBRA 114
2.9. THE SECOND GRADIENT OF A FUNCTION MAY BE BETTER THAN THE FIRST ONE
115 2.10. COUNTEREXAMPLE TO THE GENERALIZED POINCARE INEQUALITY 116
2.11. COUNTEREXAMPLE TO THE SHARPENED FRIEDRICHS INEQUALITY 121 2.12.
PLANAR DOMAINS IN EV P WHICH ARE NOT QUASIDISKS 124 2.13. COUNTEREXAMPLE
TO THE STRONG CAPACITARY INEQUALITY FOR THE NORM IN L|(FT) 129 EXERCISES
FOR CHAPTER 2 136 COMMENTS TO CHAPTER 2 137 II. SOBOLEV SPACES FOR
DOMAINS DEPENDING ON PARAMETERS 3. EXTENSION OF FUNCTIONS DEFINED ON
PARAMETER DEPENDENT DOMAINS 143 INTRODUCTION 143 3.1. ESTIMATES FOR THE
NORM OF AN EXTENSION OPERATOR TO THE EXTERIOR AND INTERIOR OF A SMALL
DOMAIN 145 XIV TABLE OF CONTENTS 3.1.1. GENERALIZED POINCARE INEQUALITY
FOR DOMAINS IN EV PL ... 145 3.1.2. AN EXTENSION FROM A SMALL DOMAIN TO
ANOTHER ONE 146 3.1.3. THE INTERIOR OF A SMALL DOMAIN 148 3.1.4.
INEQUALITIES FOR FUNCTIONS DEFINED ON A BALL 149 3.1.5. THE EXTERIOR OF
A SMALL DOMAIN 150 3.2. EXTENSION WITH ZERO BOUNDARY CONDITIONS 156 3.3.
ON THE BEST EXTENSION OPERATOR FROM A SMALL DOMAIN 162 3.4. THE
INTERIOR OF A THIN CYLINDER 168 3.4.1. AN EXTENSION OPERATOR WITH
UNIFORMLY BOUNDED NORM 169 3.4.2. THE CASE N = 1 172 3.5. A
MOLLIFICATION OPERATOR 175 3.6. EXTENSION TO THE EXTERIOR OF A THIN
CYLINDER 183 3.6.1. THREE LEMMAS ON FUNCTIONS DEFINED IN A THIN CYLINDER
184 3.6.2. AN EXTENSION OPERATOR FROM A THIN CYLINDER 190 3.7. EXTENSION
OPERATORS FOR PARTICULAR DOMAINS 194 3.7.1. EXAMPLES OF EXTENSION
OPERATORS FOR DOMAINS DEPENDING ON A SMALL PARAMETER 194 3.7.2.
EXTENSION FROM A DOMAIN DEPENDING ON TWO SMALL PARAMETERS 200 COMMENTS
TO CHAPTER 3 ;. 205 4. BOUNDARY VALUES OF FUNCTIONS S^YITH FIRST
DERIVATIVES IN L P ON PARAMETER DEPENDENT DOMAINS 207 INTRODUCTION 207
4.1. TRACES ON SMALL AND LARGE COMPONENTS OF A BOUNDARY 209 4.1.1.
GAGLIARDO S THEOREM AND ITS CONSEQUENCES 209 4.1.2. THE INTERIOR OF A
SMALL AND LARGE DOMAIN 211 4.1.3. THE EXTERIOR OF A SMALL DOMAIN 213
4.2. ON THE TRACE SPACE FOR A NARROW CYLINDER 219 4.2.1. AN EXPLICIT
NORM IN THE TRACE SPACE FOR A NARROW CYLINDER 219 4.2.2. EQUIVALENT
SEMINORMS 225 TABLE OF CONTENTS XV 4.2.3. TRACES ON THE BOUNDARY OF AN
INFINITE FUNNEL 228 4.3. INEQUALITIES FOR FUNCTIONS DEFINED ON A
CYLINDRICAL SURFACE 233 4.4. A NORM IN THE SPACE TW P FOR THE EXTERIOR
OF AN N-DIMENSIONAL CYLINDER, P N * 1 238 4.5. THE EXTERIOR OF A
CYLINDER, P N - 1 244 4.6. AN E-DEPENDENT NORM IN THE SPACE TW P FOR
THE EXTERIOR OF A CYLINDER OF WIDTH E, P = N - 1 251 COMMENTS TO CHAPTER
4 259 III. SOBOLEV SPACES FOR DOMAINS WITH CUSPS 5. EXTENSION OF
FUNCTIONS TO THE EXTERIOR OF A DOMAIN WITH THE VERTEX OF A PEAK ON THE
BOUNDARY 263 INTRODUCTION 263 5.1. INTEGRAL INEQUALITIES FOR FUNCTIONS
ON DOMAINS WITH PEAKS 265 5.1.1. FRIEDRICHS INEQUALITY FOR FUNCTIONS ON
A DOMAIN WITH OUTER PEAK 266 5.1.2. HARDY S INEQUALITIES IN DOMAINS WITH
OUTER PEAKS 267 5.2. OUTER PEAK. EXTENSION OPERATOR: V PL (Q.) - VP
JCT (R N ), IP N - 1 271 5.3. THE CASE IP = N - 1 275 5.3.1.
POSITIVE HOMOGENEOUS FUNCTIONS OF DEGREE ZERO AS MULTIPLIERS IN THE
SPACE V PL I (R N ) 275 5.3.2. LEMMA ON DIFFERENTIATION OF A CUT-OFF
FUNCTION 277 5.3.3. EXTENSION OPERATOR: V PL {Q) -» V P A (R N ), IP -
N - 1 .... 279 5.4. OUTER PEAK. EXTENSION FOR IP N - 1 285 5.4.1.
EXTENSION FROM A PEAK TO A CIRCULAR PEAK AND TO A CONE 285 5.4.2.
EXTENSION OPERATOR: V PL (Q) - V^R ) FOR IP N - 1 292 5.5. INNER
PEAKS . 297 XVI TABLE OF CONTENTS 5.5.1. THE CASE N 2 297 5.5.2.
PLANAR DOMAINS WITH INNER PEAKS 298 5.6. EXTENSION OPERATOR: VP (FT) -
V QL (K N ), Q 304 5.6.1. OUTER PEAK, THE CASE IQ N - 1 304 5.6.2.
EXTENSION OPERATOR: V^ (FT) - * V QL (R N ), IQ = N - 1 308 5.6.3. THE
CASE IQ N - 1 311 5.6.4. INNER PEAK, THE CASE N = 2 315 5.7. SMALL
PERTURBATIONS OF PEAKS IN THE VICINITY OF THE VERTEX 318 5.7.1.
TRUNCATED OUTER PEAK. EXTENSION OPERATORS: VP (FT E )) - F P (R N )
318 5.7.2. INNER TRUNCATED PEAKS, N = 2 322 COMMENTS TO CHAPTER 5 325 6.
BOUNDARY VALUES OF SOBOLEV FUNCTIONS ON NON-LIPSCHITZ DOMAINS BOUNDED BY
LIPSCHITZ SURFACES 327 INTRODUCTION 327 6.1. BALL COVERINGS OF AN OPEN
SET ASSOCIATED WITH A LIPSCHITZ FUNCTION 329 6.2. DOMAINS BETWEEN TWO
LIPSCHITZ GRAPHS 334 6.2.1. DESCRIPTION OF DOMAINS AND APPROXIMATION
LEMMA .... 334 6.2.2. TRACE THEOREMS FOR DOMAINS BETWEEN TWO LIPSCHITZ
GRAPHS 337 6.3. THE SPACE TW^FT) FOR A PLANAR DOMAIN WITH ZERO ANGLE
.... 346 - 6.4. TRACES OF FUNCTIONS IN WP (FT) FOR DOMAINS COMPLEMENTARY
TO THOSE BETWEEN LIPSCHITZ GRAPHS 352 6.5. A PLANAR DOMAIN WITH THE
VERTEX OF AN INNER PEAK ON THE BOUNDARY 359 COMMENTS TO CHAPTER 6 362 7.
BOUNDARY VALUES OF FUNCTIONS IN SOBOLEV SPACES FOR DOMAINS WITH PEAKS
363 INTRODUCTION 363 7.1. TRACES OF FUNCTIONS WITH GRADIENT IN LJ 365
TABLE OF CONTENTS XVII 7.1.1. OUTER PEAKS * 365 7.1.2. INNER PEAKS 372
7.2. THE SPACE TW^TI), P 1, FOR A DOMAIN WITH OUTER PEAK .... 374 7.3.
BOUNDARY VALUES OF FUNCTIONS IN W*(L) FOR A DOMAIN FT C R WITH INNER
PEAK, P E (1, N - 1) 381 7.4. INNER PEAK, THE CASE P = N - 1 385 7.4.1.
EQUIVALENT NORMS FOR FUNCTIONS DEFINED ON 9FT IN THE VICINITY OF THE
VERTEX OF A PEAK 386 7.4.2. TRACE THEOREM 389 7.5. APPLICATION TO THE
DIRICHLET PROBLEM FOR SECOND ORDER ELLIPTIC EQUATIONS 392 7.6.
INEQUALITIES FOR FUNCTIONS DEFINED ON A SURFACE WITH CUSP .... 394 7.7.
THE SPACE TW^Q.) FOR A DOMAIN WITH INNER PEAK, P N - 1 400 COMMENTS TO
CHAPTER 7 408 8. IMBEDDING AND TRACE THEOREMS FOR DOMAINS WITH OUTER
PEAKS AND FOR GENERAL DOMAINS 409 INTRODUCTION 409 8.1. LEMMA ON
AVERAGED FUNCTIONS 411 8.2. CONTINUITY OF THE IMBEDDING OPERATOR: V PL
(CL) - L,(FT) FOR DOMAINS WITH OUTER PEAKS 416 8.2.1. SMOOTHING OF THE
FUNCTION^DESCRIBING A CUSP 416 8.2.2. SUMMABILITY AND CONTINUITY OF
FUNCTIONS IN SOBOLEV SPACES ON DOMAINS WITH OUTER PEAKS 417 8.2.3.
IMBEDDING INTO A WEIGHTED L Q 425 8.3. COMPACTNESS THEOREM 425 8.3.1.
CRITERIA FOR COMPACTNESS OF THE IMBEDDING OPERATORS: VP (FT) - L,(FT)
AND VP (FT) - C(FT) N L TO (FT) FOR A DOMAIN WITH OUTER PEAK 425 8.3.2.
ON THE NEUMANN PROBLEM FOR A DOMAIN WITH OUTER PEAK 430 8.4. IMBEDDING
THEOREMS FOR PERTURBED PEAKS 431 XVIII TABLE OF CONTENTS 8.4.1.
TRUNCATED PEAKS 432 8.4.2. UNION OF A PEAK AND A SMALL BALL 434 8.5.
CAPACITARY CRITERIA FOR THE CONTINUITY OF THE TRACE OPERATOR: L*(FT) -
L G (FT, P) 441 8.5.1. THREE LEMMAS 441 8.5.2. CAPACITARY ISOPERIMETRIC
INEQUALITY AS A CRITERION FOR THE CONTINUITY OF THE TRACE OPERATOR:
L*(FT) -»* L G (FT, N), Q P 443 8.5.3. THE CASE Q P 444 8.6.
COMPACTNESS OF THE TRACE OPERATOR: L*(FT) - L G (FT, FI) 449 8.6.1.
COMPACTNESS OF CONTINUOUS CONVOLUTION OPERATORS L P 3FI-+K*F* L G (/Z),
Q P 450 8.6.2. THE EQUIVALENCE OF THE CONTINUITY AND COMPACTNESS OF
THE TRACE OPERATOR: L*(FT) - - L Q (FT, /Z), Q P, P 1 ... 452 8.6.3.
COMPACTNESS THEOREM IN THE CASE Q P 455 COMMENTS TO CHAPTER 8 458
REFERENCES 461 INDEX 477 LIST OF SYMBOLS 481
|
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| id | DE-604.BV013365648 |
| illustrated | Illustrated |
| indexdate | 2024-07-09T18:44:34Z |
| institution | BVB |
| isbn | 9810227671 |
| language | English |
| oai_aleph_id | oai:aleph.bib-bvb.de:BVB01-009116835 |
| oclc_num | 247053144 |
| open_access_boolean | |
| owner | DE-19 DE-BY-UBM DE-11 DE-188 DE-384 |
| owner_facet | DE-19 DE-BY-UBM DE-11 DE-188 DE-384 |
| physical | XVIII, 481 S. graph. Darst. |
| publishDate | 1997 |
| publishDateSearch | 1997 |
| publishDateSort | 1997 |
| publisher | World Scientific |
| record_format | marc |
| spelling | Mazʹja, Vladimir Gilelevič 1937- Verfasser (DE-588)121490602 aut Differentiable functions on bad domains Vladimir G. Maz'ya ; Sergei V. Poborchi Singapore [u.a.] World Scientific 1997 XVIII, 481 S. graph. Darst. txt rdacontent n rdamedia nc rdacarrier Sobolev-Raum - Differenzierbare Funktion Sobolev-Raum (DE-588)4055345-0 gnd rswk-swf Differenzierbare Funktion (DE-588)4149803-3 gnd rswk-swf Sobolev-Raum (DE-588)4055345-0 s Differenzierbare Funktion (DE-588)4149803-3 s DE-604 Poborchi, Sergei V. Verfasser aut GBV Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=009116835&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Mazʹja, Vladimir Gilelevič 1937- Poborchi, Sergei V. Differentiable functions on bad domains Sobolev-Raum - Differenzierbare Funktion Sobolev-Raum (DE-588)4055345-0 gnd Differenzierbare Funktion (DE-588)4149803-3 gnd |
| subject_GND | (DE-588)4055345-0 (DE-588)4149803-3 |
| title | Differentiable functions on bad domains |
| title_auth | Differentiable functions on bad domains |
| title_exact_search | Differentiable functions on bad domains |
| title_full | Differentiable functions on bad domains Vladimir G. Maz'ya ; Sergei V. Poborchi |
| title_fullStr | Differentiable functions on bad domains Vladimir G. Maz'ya ; Sergei V. Poborchi |
| title_full_unstemmed | Differentiable functions on bad domains Vladimir G. Maz'ya ; Sergei V. Poborchi |
| title_short | Differentiable functions on bad domains |
| title_sort | differentiable functions on bad domains |
| topic | Sobolev-Raum - Differenzierbare Funktion Sobolev-Raum (DE-588)4055345-0 gnd Differenzierbare Funktion (DE-588)4149803-3 gnd |
| topic_facet | Sobolev-Raum - Differenzierbare Funktion Sobolev-Raum Differenzierbare Funktion |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=009116835&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
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