Linear second order elliptic operators:
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| Main Author: | |
|---|---|
| Format: | Book |
| Language: | English |
| Published: |
New Jersey [u.a.]
World Scientific
2013
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| Subjects: | |
| Online Access: | Inhaltsverzeichnis |
| Physical Description: | XVIII, 337 S. graph. Darst. |
| ISBN: | 9789814440240 |
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| 100 | 1 | |a López-Gómez, Julián |e Verfasser |4 aut | |
| 245 | 1 | 0 | |a Linear second order elliptic operators |c Julián López-Gómez |
| 264 | 1 | |a New Jersey [u.a.] |b World Scientific |c 2013 | |
| 300 | |a XVIII, 337 S. |b graph. Darst. | ||
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| 999 | |a oai:aleph.bib-bvb.de:BVB01-026129248 | ||
Record in the Search Index
| _version_ | 1804150557165748224 |
|---|---|
| adam_text | Titel: Linear second order elliptic operators
Autor: López-Gómez, Julián
Jahr: 2013
Contents
Preface vii
1. The minimum principle 1
1.1 Concept of ellipticity. First consequences ......... 2
1.2 Minimum principle of E. Hopf................ 5
1.3 Interior sphere properties.................. 12
1.4 Boundary lemma of E. Hopf................. 19
1.5 Positivity properties of super-harmonic functions..... 23
1.6 Uniform decay property of E. Hopf............. 25
1.7 The generalized minimum principle of M. H. Protter and
H. F. Weinberger....................... 30
1.8 Appendix: Smooth domains................. 32
1.9 Comments on Chapter 1................... 38
2. Classifying supersolutions 41
2.1 First classification theorem ................. 42
2.2 Existence of positive strict supersolutions......... 47
2.3 Positivity of the resolvent operator............. 52
2.4 Behavior of the positive supersolutions on To....... 52
2.5 Second classification theorem................ 53
2.6 Appendix: Partitions of the unity.............. 58
2.7 Comments on Chapter 2................... 60
3. Representation theorems 63
3.1 The projection on a closed convex set ........... 65
3.2 The orthogonal projection on a closed subspace...... 69
3.3 The representation theorem of F. Riesz .......... 71
3.4 Continuity and coercivity of bilinear forms......... 75
3.5 The theorem of G. Stampacchia............... 76
3.6 The theorem of P. D. Lax and A. N. Milgram....... 78
3.7 Projecting on a closed convex set of a u.c. B-space .... 78
3.7.1 Basic concepts and preliminaries.......... 79
3.7.2 The projection theorem............... 82
3.7.3 The projection on a closed linear subspace .... 85
3.7.4 The projection on a closed hyperplane ...... 87
3.8 Comments on Chapter 3................... 88
4. Existence of weak solutions 91
4.1 Preliminaries. Sobolev spaces................ 93
4.1.1 Test functions.................... 93
4.1.2 Weak derivatives. Sobolev spaces ......... 94
4.1.3 Holder spaces of continuous functions....... 97
4.1.4 Sobolev s imbeddings................ 98
4.1.5 Compact imbeddings................ 101
4.2 Trace operators........................ 102
4.3 Weak solutions........................ 114
4.4 Continuity of the associated bilinear form......... 117
4.5 Invertibility of (4.4) when ß 0 .............. 118
4.5.1 Coercivity of the associated bilinear form..... 118
4.5.2 Existence of weak solutions. The resolvent operator 120
4.6 Invertibility of (4.4) for arbitrary ß............. 122
4.7 Comments on Chapter 4................... 126
5. Regularity of weak solutions 129
5.1 Lp(RN)-estimates for the Laplacian ............ 131
5.2 Zp( )-estimates for the Laplacian ............. 135
5.3 General elliptic Lp( )-estimates when T1 = 0....... 138
5.4 The method of continuity.................. 139
5.5 Regularity of weak solutions when T1 = 0......... 141
5.6 A first glance to the general case when T1 = 0....... 147
5.7 Comments on Chapter 5................... 152
6. The Krein-Rutman theorem 155
6.1 Orderings. Ordered Banach spaces............. 155
6.2 Spectral theory of linear compact operators........ 161
6.3 The Krein-Rutman theorem................. 164
6.4 Preliminaries of the proof of Theorem 6.3......... 166
6.5 Proof of Theorem 6.3..................... 171
6.6 Comments on Chapter 6................... 184
7. The strong maximum principle 187
7.1 Minimum principle of J. M. Bony.............. 189
7.2 The existence of the principal eigenvalue.......... 195
7.3 Two equivalent weak eigenvalue problems......... 206
7.4 Simplicity and dominance of o[L, B, ]........... 208
7.4.1 Proof of the strict dominance in case To = 0 ... 210
7.4.2 Proof of the strict dominance in case T1 = 0 ... 212
7.4.3 Proof of the strict dominance in the general case . 215
7.5 The strong maximum principle............... 215
7.6 The classical minimum principles revisited......... 217
7.7 Comments on Chapter 7................... 220
8. Properties of the principal eigenvalue 225
8.1 Monotonicity properties................... 226
8.2 Point-wise min-max characterizations ........... 229
8.3 Concavity with respect to the potential .......... 232
8.4 Stability of along the Dirichlet components of O ... 234
8.4.1 Proof of Proposition 8.5 .............. 236
8.4.2 Proof of Theorem 8.4................ 240
8.5 Continuous dependence with respect to ......... 240
8.6 Continuous dependence with respect to ß(x) ....... 254
8.7 Asymptotic behavior of o (ß) as min ß f oo......... 260
8.8 Lower estimates of o [L,D, ] in terms of | |........ 264
8.9 Comments on Chapter 8................... 267
9. Principal eigenvalues of linear weighted boundary value problems 273
9.1 General properties of the map (A)............. 274
9.2 Characterizing the existence of a principal eigenvalue . . . 278
9.3 Ascertaining limA-00o[L + XV,B, ] when V 0..... 284
9.3.1 The simplest case.................. 285
9.3.2 The admissible V s satisfying the main theorem . 287
9.3.3 The main theorem.................. 290
9.4 Characterizing the existence of principal eigenvalues for
admissible potentials..................... 307
9.5 Comments on Chapter 9................... 311
Bibliography 319
Index 331
|
| any_adam_object | 1 |
| author | López-Gómez, Julián |
| author_facet | López-Gómez, Julián |
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| discipline | Mathematik |
| format | Book |
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| id | DE-604.BV041153835 |
| illustrated | Illustrated |
| indexdate | 2024-07-10T00:40:49Z |
| institution | BVB |
| isbn | 9789814440240 |
| language | English |
| lccn | 2013001024 |
| oai_aleph_id | oai:aleph.bib-bvb.de:BVB01-026129248 |
| oclc_num | 855544857 |
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| owner | DE-703 DE-91G DE-BY-TUM |
| owner_facet | DE-703 DE-91G DE-BY-TUM |
| physical | XVIII, 337 S. graph. Darst. |
| publishDate | 2013 |
| publishDateSearch | 2013 |
| publishDateSort | 2013 |
| publisher | World Scientific |
| record_format | marc |
| spelling | López-Gómez, Julián Verfasser aut Linear second order elliptic operators Julián López-Gómez New Jersey [u.a.] World Scientific 2013 XVIII, 337 S. graph. Darst. txt rdacontent n rdamedia nc rdacarrier Elliptic operators Elliptischer Differentialoperator (DE-588)4140057-4 gnd rswk-swf Maximumprinzip (DE-588)4169165-9 gnd rswk-swf Elliptischer Differentialoperator (DE-588)4140057-4 s Maximumprinzip (DE-588)4169165-9 s DE-604 HBZ Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=026129248&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | López-Gómez, Julián Linear second order elliptic operators Elliptic operators Elliptischer Differentialoperator (DE-588)4140057-4 gnd Maximumprinzip (DE-588)4169165-9 gnd |
| subject_GND | (DE-588)4140057-4 (DE-588)4169165-9 |
| title | Linear second order elliptic operators |
| title_auth | Linear second order elliptic operators |
| title_exact_search | Linear second order elliptic operators |
| title_full | Linear second order elliptic operators Julián López-Gómez |
| title_fullStr | Linear second order elliptic operators Julián López-Gómez |
| title_full_unstemmed | Linear second order elliptic operators Julián López-Gómez |
| title_short | Linear second order elliptic operators |
| title_sort | linear second order elliptic operators |
| topic | Elliptic operators Elliptischer Differentialoperator (DE-588)4140057-4 gnd Maximumprinzip (DE-588)4169165-9 gnd |
| topic_facet | Elliptic operators Elliptischer Differentialoperator Maximumprinzip |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=026129248&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
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