Hörmander spaces, interpolation, and elliptic problems:
Saved in:
| Main Authors: | , |
|---|---|
| Format: | Book |
| Language: | English |
| Published: |
Berlin [u.a.]
De Gruyter
2014
|
| Series: | De Gruyter studies in mathematics
60 |
| Subjects: | |
| Online Access: | Inhaltstext Inhaltsverzeichnis Klappentext |
| Physical Description: | XII, 297 S. graph. Darst. 240 mm x 170 mm |
| ISBN: | 3110296853 9783110296853 |
Staff View
MARC
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| 100 | 1 | |a Mikhailets, Vladimir A. |e Verfasser |4 aut | |
| 245 | 1 | 0 | |a Hörmander spaces, interpolation, and elliptic problems |c Vladimir A. Mikhailets ; Aleksandr A. Murach |
| 264 | 1 | |a Berlin [u.a.] |b De Gruyter |c 2014 | |
| 300 | |a XII, 297 S. |b graph. Darst. |c 240 mm x 170 mm | ||
| 336 | |b txt |2 rdacontent | ||
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| 650 | 0 | 7 | |a Elliptisches System |0 (DE-588)4121184-4 |2 gnd |9 rswk-swf |
| 689 | 0 | 0 | |a Elliptischer Differentialoperator |0 (DE-588)4140057-4 |D s |
| 689 | 0 | 1 | |a Elliptisches Randwertproblem |0 (DE-588)4193399-0 |D s |
| 689 | 0 | 2 | |a Elliptisches System |0 (DE-588)4121184-4 |D s |
| 689 | 0 | |5 DE-604 | |
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CONTENTS
PREFACE ' V
PREFACE TO THE ENGLISH EDITION VI
ACKNOWLEDGEMENTS
VII
INTRODUCTION 1
1 INTERPOLATION AND HORMANDER SPACES 9
1.1 INTERPOLATION WITH FUNCTION PARAMETER 9
1.1.1 DEFINITION OF INTERPOLATION 9
1.1.2 EMBEDDINGS OF SPACES 11
1.1.3 REITERATION PROPERTY 13
1.1.4 INTERPOLATION OF DUAL SPACES 15
1.1.5 INTERPOLATION OF ORTHOGONAL SUMS OF SPACES 18
1.1.6 INTERPOLATION OF SUBSPACES AND FACTOR SPACES 20
1.1.7 INTERPOLATION OF FREDHOLM OPERATORS 21
1.1.8 ESTIMATE OF THE OPERATOR NORM IN INTERPOLATION SPACES . 23
1.1.9 CRITERION FOR A FUNCTION TO BE AN INTERPOLATION PARAMETER 25
1.2 REGULARLY VARYING FUNCTIONS AND THEIR GENERALIZATION 29
1.2.1 REGULARLY VARYING FUNCTIONS 29
1.2.2 QUASIREGULARLY VARYING FUNCTIONS 31
1.2.3 AUXILIARY RESULTS 36
1.3 HORMANDER SPACES AND THE REFINED SOBOLEV SCALE 38
1.3.1 PRELIMINARY INFORMATION AND NOTATION 38
1.3.2 HORMANDER SPACES 40
1.3.3 REFINED SOBOLEV SCALE 42
1.3.4 PROPERTIES OF THE REFINED SCALE 44
1.4 UNIFORMLY ELLIPTIC OPERATORS ON THE REFINED SCALE 47
1.4.1 PSEUDODIFFERENTIAL OPERATORS 47
1.4.2 A PRIORI ESTIMATE OF THE SOLUTIONS 50
1.4.3 SMOOTHNESS OF THE SOLUTIONS 51
1.5 REMARKS AND COMMENTS 55
2 HORMANDER SPACES ON CLOSED MANIFOLDS AND THEIR APPLICATIONS 59
2.1 HORMANDER SPACES ON CLOSED MANIFOLDS 59
2.1.1 EQUIVALENT DEFINITIONS 59
HTTP://D-NB.INFO/1048116506
X
CONTENTS
2.1.2 INTERPOLATION PROPERTIES 61
2.1.3 EQUIVALENT NORMS 68
2.1.4 EMBEDDING THEOREM 77
2.2 ELLIPTIC OPERATORS ON CLOSED MANIFOLDS 78
2.2.1 PSEUDODIFFERENTIAL OPERATORS ON CLOSED MANIFOLDS 79
2.2.2 ELLIPTIC OPERATORS ON THE REFINED SCALE 81
2.2.3 SMOOTHNESS OF SOLUTIONS TO THE ELLIPTIC EQUATION 84
2.2.4 PARAMETER-ELLIPTIC OPERATORS 86
2.3 CONVERGENCE OF SPECTRAL EXPANSIONS 93
2.3.1 CONVERGENCE ALMOST EVERYWHERE FOR GENERAL ORTHOGONAL
SERIES 93
2.3.2 CONVERGENCE ALMOST EVERYWHERE FOR SPECTRAL EXPANSIONS . 95
2.3.3 CONVERGENCE OF SPECTRAL EXPANSIONS IN THE METRIC OF THE
SPACE C
K
97
2.4 RO-VARYING FUNCTIONS AND HORMANDER SPACES 98
2.4.1 RO-VARYING FUNCTIONS IN THE SENSE OF AVAKUMOVIC 98
2.4.2 INTERPOLATION SPACES FOR A PAIR OF SOBOLEV SPACES 100
2.4.3 APPLICATIONS TO ELLIPTIC OPERATORS 107
2.5 REMARKS AND COMMENTS 108
3 SEMIHOMOGENEOUS ELLIPTIC BOUNDARY-VALUE PROBLEMS 111
3.1 REGULAR ELLIPTIC BOUNDARY-VALUE PROBLEMS ILL
3.1.1 DEFINITION OF THE PROBLEM ILL
3.1.2 FORMALLY ADJOINT PROBLEM 113
3.2 HORMANDER SPACES FOR EUCLIDEAN DOMAINS 114
3.2.1 SPACES FOR OPEN DOMAINS 115
3.2.2 SPACES FOR CLOSED DOMAINS 120
3.2.3 RIGGING OF L%(1)
WITH HORMANDER SPACES 123
3.3 BOUNDARY-VALUE PROBLEMS FOR HOMOGENEOUS ELLIPTIC EQUATIONS . . . 126
3.3.1 MAIN RESULT: BOUNDEDNESS AND FREDHOLM PROPERTY OF THE
OPERATOR 126
3.3.2 A THEOREM ON INTERPOLATION OF SUBSPACES 127
3.3.3 ELLIPTIC BOUNDARY-VALUE PROBLEM IN SOBOLEV SPACES 131
3.3.4 PROOF OF THE MAIN RESULT 134
3.3.5 PROPERTIES OF SOLUTIONS TO THE HOMOGENEOUS ELLIPTIC
EQUATION 139
3.4 ELLIPTIC PROBLEMS WITH HOMOGENEOUS BOUNDARY CONDITIONS 142
3.4.1 THEOREM ON ISOMORPHISMS FOR ELLIPTIC OPERATORS 142
3.4.2 INTERPOLATION AND HOMOGENEOUS BOUNDARY CONDITIONS . 146
CONTENTS XI
3.4.3 PROOFS OF THEOREMS ON ISOMORPHISMS AND THE FREDHOLM
PROPERTY 152
3.4.4 LOCAL INCREASE IN SMOOTHNESS OF SOLUTIONS UP TO THE
BOUNDARY 156
3.5 SOME PROPERTIES OF HORMANDER SPACES 158
3.5.1 SPACE
H
Q
V
(
Q
.) AND ITS PROPERTIES 158
3.5.2 EQUIVALENT DESCRIPTION OF
H
3: FI
(Q.)
160
3.6 REMARKS AND COMMENTS 162
4 INHOMOGENEOUS ELLIPTIC BOUNDARY-VALUE PROBLEMS 165
4.1 ELLIPTIC BOUNDARY-VALUE PROBLEMS IN THE POSITIVE ONE-SIDED SCALE 165
4.1.1 THEOREMS ON FREDHOLM PROPERTY AND ISOMORPHISMS 165
4.1.2 SMOOTHNESS OF THE SOLUTIONS UP TO THE BOUNDARY 169
4.1.3 NONREGULAR ELLIPTIC BOUNDARY-VALUE PROBLEMS 173
4.1.4 PARAMETER-ELLIPTIC BOUNDARY-VALUE PROBLEMS 175
4.1.5 FORMALLY MIXED ELLIPTIC BOUNDARY-VALUE PROBLEM 186
4.2 ELLIPTIC BOUNDARY-VALUE PROBLEMS IN THE TWO-SIDED SCALE 188
4.2.1 PRELIMINARY REMARKS 188
4.2.2 THE REFINED SCALE MODIFIED IN THE SENSE OF ROITBERG 189
4.2.3 ROITBERG-TYPE THEOREMS ON SOLVABILITY. THE COMPLETE
COLLECTION OF ISOMORPHISMS 199
4.2.4 SMOOTHNESS OF GENERALIZED SOLUTIONS UP TO THE BOUNDARY . 204
4.2.5 INTERPOLATION IN THE MODIFIED REFINED SCALE 207
4.3 SOME PROPERTIES OF THE MODIFIED REFINED SCALE 210
4.3.1 STATEMENT OF RESULTS 210
4.3.2 PROOF OF RESULTS 212
4.4 GENERALIZATION OF THE LIONS-MAGENES THEOREMS 226
4.4.1 LIONS-MAGENES THEOREMS 227
4.4.2 KEY INDIVIDUAL THEOREM 230
4.4.3 INDIVIDUAL THEOREM FOR SOBOLEV SPACES 236
4.4.4 INDIVIDUAL THEOREM FOR WEIGHT SPACES 238
4.5 HORMANDER SPACES AND INDIVIDUAL THEOREMS ON SOLVABILITY 243
4.5.1 KEY INDIVIDUAL THEOREM FOR THE REFINED SCALE 243
4.5.2 OTHER INDIVIDUAL THEOREMS 244
4.6 REMARKS AND COMMENTS 247
5 ELLIPTIC SYSTEMS 251
5.1 UNIFORMLY ELLIPTIC SYSTEMS IN THE REFINED SOBOLEV SCALE 251
5.1.1 UNIFORMLY ELLIPTIC SYSTEMS 251
5.1.2 A PRIORI ESTIMATE FOR THE SOLUTIONS OF THE SYSTEM 252
5.1.3 SMOOTHNESS OF SOLUTIONS 253
XII CONTENTS
5.2 ELLIPTIC SYSTEMS ON A CLOSED MANIFOLD 257
5.2.1 ELLIPTIC SYSTEMS 257
5.2.2 OPERATOR OF THE ELLIPTIC SYSTEM ON THE REFINED SCALE 258
5.2.3 LOCAL SMOOTHNESS OF SOLUTIONS 262
5.2.4 PARAMETER-ELLIPTIC SYSTEMS 264
5.3 ELLIPTIC BOUNDARY-VALUE PROBLEMS FOR SYSTEMS OF EQUATIONS 268
5.3.1 STATEMENT OF THE PROBLEM 269
5.3.2 THEOREM ON SOLVABILITY 271
5.4 REMARKS AND COMMENTS 272
BIBLIOGRAPHY 275
INDEX 291
The monograph gives a detailed exposition of the theory of general elliptic operators
(scalar and matrix) and elliptic boundary value problems in Hubert scales of
Hörmander
function spaces. This theory was constructed by the authors in a number
of papers published in
2005-2009.
It is distinguished by a systematic use of the method
of interpolation with a functional parameter of abstract Hubert spaces and Sobolev
inner product spaces. This method, the theory and their applications are expounded
for the first time in the monographic literature. The monograph is written in detail and
in a reader-friendly style. The complete proofs of theorems are given, This monograph
is intended for a wide range of mathematicians whose research interests concern with
mathematical analysis and differential equations, |
| any_adam_object | 1 |
| author | Mikhailets, Vladimir A. Murach, Aleksandr A. |
| author_facet | Mikhailets, Vladimir A. Murach, Aleksandr A. |
| author_role | aut aut |
| author_sort | Mikhailets, Vladimir A. |
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| bvnumber | BV041814642 |
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| ctrlnum | (OCoLC)872699548 (DE-599)DNB1048116506 |
| dewey-full | 515.3533 |
| dewey-hundreds | 500 - Natural sciences and mathematics |
| dewey-ones | 515 - Analysis |
| dewey-raw | 515.3533 |
| dewey-search | 515.3533 |
| dewey-sort | 3515.3533 |
| dewey-tens | 510 - Mathematics |
| discipline | Mathematik |
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| id | DE-604.BV041814642 |
| illustrated | Illustrated |
| indexdate | 2024-09-10T01:12:38Z |
| institution | BVB |
| isbn | 3110296853 9783110296853 |
| language | English |
| oai_aleph_id | oai:aleph.bib-bvb.de:BVB01-027259916 |
| oclc_num | 872699548 |
| open_access_boolean | |
| owner | DE-20 DE-19 DE-BY-UBM DE-29T DE-11 DE-739 DE-634 |
| owner_facet | DE-20 DE-19 DE-BY-UBM DE-29T DE-11 DE-739 DE-634 |
| physical | XII, 297 S. graph. Darst. 240 mm x 170 mm |
| publishDate | 2014 |
| publishDateSearch | 2014 |
| publishDateSort | 2014 |
| publisher | De Gruyter |
| record_format | marc |
| series | De Gruyter studies in mathematics |
| series2 | De Gruyter studies in mathematics |
| spelling | Mikhailets, Vladimir A. Verfasser aut Hörmander spaces, interpolation, and elliptic problems Vladimir A. Mikhailets ; Aleksandr A. Murach Berlin [u.a.] De Gruyter 2014 XII, 297 S. graph. Darst. 240 mm x 170 mm txt rdacontent n rdamedia nc rdacarrier De Gruyter studies in mathematics 60 Elliptisches Randwertproblem (DE-588)4193399-0 gnd rswk-swf Elliptischer Differentialoperator (DE-588)4140057-4 gnd rswk-swf Elliptisches System (DE-588)4121184-4 gnd rswk-swf Elliptischer Differentialoperator (DE-588)4140057-4 s Elliptisches Randwertproblem (DE-588)4193399-0 s Elliptisches System (DE-588)4121184-4 s DE-604 Murach, Aleksandr A. Verfasser aut Erscheint auch als Online-Ausgabe 978-3-11-029689-1 De Gruyter studies in mathematics 60 (DE-604)BV000005407 60 X:MVB text/html http://deposit.dnb.de/cgi-bin/dokserv?id=4607427&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=027259916&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis Digitalisierung UB Passau - ADAM Catalogue Enrichment application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=027259916&sequence=000004&line_number=0002&func_code=DB_RECORDS&service_type=MEDIA Klappentext |
| spellingShingle | Mikhailets, Vladimir A. Murach, Aleksandr A. Hörmander spaces, interpolation, and elliptic problems De Gruyter studies in mathematics Elliptisches Randwertproblem (DE-588)4193399-0 gnd Elliptischer Differentialoperator (DE-588)4140057-4 gnd Elliptisches System (DE-588)4121184-4 gnd |
| subject_GND | (DE-588)4193399-0 (DE-588)4140057-4 (DE-588)4121184-4 |
| title | Hörmander spaces, interpolation, and elliptic problems |
| title_auth | Hörmander spaces, interpolation, and elliptic problems |
| title_exact_search | Hörmander spaces, interpolation, and elliptic problems |
| title_full | Hörmander spaces, interpolation, and elliptic problems Vladimir A. Mikhailets ; Aleksandr A. Murach |
| title_fullStr | Hörmander spaces, interpolation, and elliptic problems Vladimir A. Mikhailets ; Aleksandr A. Murach |
| title_full_unstemmed | Hörmander spaces, interpolation, and elliptic problems Vladimir A. Mikhailets ; Aleksandr A. Murach |
| title_short | Hörmander spaces, interpolation, and elliptic problems |
| title_sort | hormander spaces interpolation and elliptic problems |
| topic | Elliptisches Randwertproblem (DE-588)4193399-0 gnd Elliptischer Differentialoperator (DE-588)4140057-4 gnd Elliptisches System (DE-588)4121184-4 gnd |
| topic_facet | Elliptisches Randwertproblem Elliptischer Differentialoperator Elliptisches System |
| url | http://deposit.dnb.de/cgi-bin/dokserv?id=4607427&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=027259916&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=027259916&sequence=000004&line_number=0002&func_code=DB_RECORDS&service_type=MEDIA |
| volume_link | (DE-604)BV000005407 |
| work_keys_str_mv | AT mikhailetsvladimira hormanderspacesinterpolationandellipticproblems AT murachaleksandra hormanderspacesinterpolationandellipticproblems |