Variational methods for potential operator equations: with applications to nonlinear elliptic equations
Gespeichert in:
| 1. Verfasser: | |
|---|---|
| Format: | Buch |
| Sprache: | German |
| Veröffentlicht: |
Berlin [u.a.]
<<de>> Gruyter
1997
|
| Schriftenreihe: | De Gruyter studies in mathematics
24 |
| Schlagworte: | |
| Online-Zugang: | Inhaltsverzeichnis |
| Beschreibung: | Literaturverz. S. 270 - 286 |
| Beschreibung: | IX, 290 S. |
| ISBN: | 311015269X |
Internformat
MARC
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| 100 | 1 | |a Chabrowski, Jan |e Verfasser |4 aut | |
| 245 | 1 | 0 | |a Variational methods for potential operator equations |b with applications to nonlinear elliptic equations |c Jan Chabrowski |
| 264 | 1 | |a Berlin [u.a.] |b <<de>> Gruyter |c 1997 | |
| 300 | |a IX, 290 S. | ||
| 336 | |b txt |2 rdacontent | ||
| 337 | |b n |2 rdamedia | ||
| 338 | |b nc |2 rdacarrier | ||
| 490 | 1 | |a De Gruyter studies in mathematics |v 24 | |
| 500 | |a Literaturverz. S. 270 - 286 | ||
| 650 | 0 | 7 | |a Variationsproblem |0 (DE-588)4187419-5 |2 gnd |9 rswk-swf |
| 650 | 0 | 7 | |a Potenzialoperator |0 (DE-588)4175486-4 |2 gnd |9 rswk-swf |
| 650 | 0 | 7 | |a Elliptische Differentialgleichung |0 (DE-588)4014485-9 |2 gnd |9 rswk-swf |
| 689 | 0 | 0 | |a Elliptische Differentialgleichung |0 (DE-588)4014485-9 |D s |
| 689 | 0 | 1 | |a Potenzialoperator |0 (DE-588)4175486-4 |D s |
| 689 | 0 | 2 | |a Variationsproblem |0 (DE-588)4187419-5 |D s |
| 689 | 0 | |5 DE-604 | |
| 830 | 0 | |a De Gruyter studies in mathematics |v 24 |w (DE-604)BV000005407 |9 24 | |
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Datensatz im Suchindex
| _version_ | 1804125777531240448 |
|---|---|
| adam_text | Contents
1 Constrained minimization 1
1.1 Preliminaries 1
1.2 Constrained minimization 8
1.3 Dual method 13
1.4 Minimizers with the least energy 14
1.5 Application of dual method 15
1.6 Multiple solutions of nonhomogeneous equation 17
1.7 Sets of constraints 19
1.8 Constrained minimization for Ff 24
1.9 Subcritical problem 29
1.10 Application to the p Laplacian 30
1.11 Critical problem 35
1.12 Bibliographical notes 37
2 Applications of Lusternik Schnirelman theory 39
2.1 Palais Smale condition, case p ^ q 39
2.2 Duality mapping 40
2.3 Palais Smale condition, case p = q 43
2.4 The Lusternik Schnirelman theory 47
2.5 Case p q 55
2.6 Case p q 56
2.7 Case p = q 60
2.8 The p Laplacian in bounded domain 63
2.9 Iterative construction of eigenvectors 67
2.10 Critical points of higher order 70
2.11 Bibliographical notes 73
3 Nonhomogeneous potentials 74
3.1 Preliminaries and assumptions 74
3.2 Constrained minimization 76
3.3 Application — compact case 79
3.4 Perturbation theorems — noncompact case 81
3.5 Perturbation of the functional a — noncompact case 85
3.6 Existence of infinitely many solutions 88
viii Contents
3.7 General minimization — case p q 90
3.8 Set of constraints V 99
3.9 Application to a critical case p — n 101
3.10 Technical lemmas 103
3.11 Existence result for problem (3.34) 112
3.12 Bibliographical notes 113
4 Potentials with covariance condition 115
4.1 Preliminaries and constrained minimization 115
4.2 Dual method 120
4.3 Minimization subject to constraint V 120
4.4 Sobolev inequality 121
4.5 Mountain pass theorem and constrained minimization 122
4.6 Minimization problem for a system of equations 125
4.7 Bibliographical notes 127
5 Eigenvalues and level sets 128
5.1 Level sets 128
5.2 Continuity and monotonicity of a 130
5.3 The differentiability properties of a 132
5.4 Schechter s version of the mountain pass theorem 135
5.5 General condition for solvability of (5.11) 138
5.6 Properties of the function/c(r) 140
5.7 Hilbert space case 142
5.8 Application to elliptic equations 143
5.9 Bibliographical notes 148
6 Generalizations of the mountain pass theorem 149
6.1 Version of a deformation lemma 149
6.2 Mountain pass alternative 153
6.3 Consequences of mountain pass alternative 155
6.4 Hampwile alternative 157
6.5 Applicability of the mountain pass theorem 160
6.6 Mountain pass and Hampwile alternative 163
6.7 Bibliographical notes 166
7 Nondifferentiable functionals 167
7.1 Concept of a generalized gradient 167
7.2 Generalized gradients in function spaces 172
7.3 Mountain pass theorem for locally Lipschitz functionals 174
7.4 Consequences of Theorem 7.3.1 181
7.5 Application to boundary value problem with discontinuous nonlinearity 183
7.6 Lower semicontinuous perturbation 185
7.7 Deformation lemma for functionals satisfying condition (L) 188
Contents ix
7.8 Application to variational inequalities 195
7.9 Bibliographical notes 197
8 Concentration compactness principle — subcritical case 198
8.1 Concentration compactness principle at infinity — subcritical case . . 198
8.2 Constrained minimization — subcritical case 200
8.3 Constrained minimization with b ^ const, subcritical case 205
8.4 Behaviour of the Palais Smale sequences 211
8.5 The exterior Dirichlet problem 215
8.6 The Palais Smale condition 218
8.7 Concentration compactness principle I 221
8.8 Bibliographical notes 223
9 Concentration compactness principle — critical case 224
9.1 Critical Sobolev exponent 224
9.2 Concentration compactness principle II 228
9.3 Loss of mass at infinity 229
9.4 Constrained minimization — critical case 233
9.5 Palais Smale sequences in critical case 237
9.6 Symmetric solutions 244
9.7 Remarks on compact embeddings into L2*(Q) and L^*(K ) 250
9.8 Bibliographical notes 252
Appendix 253
A.I Sobolev spaces 253
A.2 Embedding theorems 254
A.3 Compact embeddings of spaces W P(Rn) and Dl P(W) 255
A.4 Conditions of concentration and uniform decay at infinity 259
A.5 Compact embedding for H*(W) 261
A.6 Schwarz symmetrization 264
A.7 Pointwise convergence 264
A.8 Gateaux derivatives 266
Bibliography 270
Glossary 287
Index 289
|
| any_adam_object | 1 |
| author | Chabrowski, Jan |
| author_facet | Chabrowski, Jan |
| author_role | aut |
| author_sort | Chabrowski, Jan |
| author_variant | j c jc |
| building | Verbundindex |
| bvnumber | BV011273103 |
| classification_rvk | SK 560 SK 620 |
| ctrlnum | (OCoLC)845374017 (DE-599)BVBBV011273103 |
| dewey-full | 515/.64 |
| dewey-hundreds | 500 - Natural sciences and mathematics |
| dewey-ones | 515 - Analysis |
| dewey-raw | 515/.64 |
| dewey-search | 515/.64 |
| dewey-sort | 3515 264 |
| dewey-tens | 510 - Mathematics |
| discipline | Mathematik |
| format | Book |
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| id | DE-604.BV011273103 |
| illustrated | Not Illustrated |
| indexdate | 2024-07-09T18:06:57Z |
| institution | BVB |
| isbn | 311015269X |
| language | German |
| oai_aleph_id | oai:aleph.bib-bvb.de:BVB01-007570180 |
| oclc_num | 845374017 |
| open_access_boolean | |
| owner | DE-824 DE-384 DE-706 DE-634 DE-188 DE-11 |
| owner_facet | DE-824 DE-384 DE-706 DE-634 DE-188 DE-11 |
| physical | IX, 290 S. |
| publishDate | 1997 |
| publishDateSearch | 1997 |
| publishDateSort | 1997 |
| publisher | <<de>> Gruyter |
| record_format | marc |
| series | De Gruyter studies in mathematics |
| series2 | De Gruyter studies in mathematics |
| spelling | Chabrowski, Jan Verfasser aut Variational methods for potential operator equations with applications to nonlinear elliptic equations Jan Chabrowski Berlin [u.a.] <<de>> Gruyter 1997 IX, 290 S. txt rdacontent n rdamedia nc rdacarrier De Gruyter studies in mathematics 24 Literaturverz. S. 270 - 286 Variationsproblem (DE-588)4187419-5 gnd rswk-swf Potenzialoperator (DE-588)4175486-4 gnd rswk-swf Elliptische Differentialgleichung (DE-588)4014485-9 gnd rswk-swf Elliptische Differentialgleichung (DE-588)4014485-9 s Potenzialoperator (DE-588)4175486-4 s Variationsproblem (DE-588)4187419-5 s DE-604 De Gruyter studies in mathematics 24 (DE-604)BV000005407 24 HBZ Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=007570180&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Chabrowski, Jan Variational methods for potential operator equations with applications to nonlinear elliptic equations De Gruyter studies in mathematics Variationsproblem (DE-588)4187419-5 gnd Potenzialoperator (DE-588)4175486-4 gnd Elliptische Differentialgleichung (DE-588)4014485-9 gnd |
| subject_GND | (DE-588)4187419-5 (DE-588)4175486-4 (DE-588)4014485-9 |
| title | Variational methods for potential operator equations with applications to nonlinear elliptic equations |
| title_auth | Variational methods for potential operator equations with applications to nonlinear elliptic equations |
| title_exact_search | Variational methods for potential operator equations with applications to nonlinear elliptic equations |
| title_full | Variational methods for potential operator equations with applications to nonlinear elliptic equations Jan Chabrowski |
| title_fullStr | Variational methods for potential operator equations with applications to nonlinear elliptic equations Jan Chabrowski |
| title_full_unstemmed | Variational methods for potential operator equations with applications to nonlinear elliptic equations Jan Chabrowski |
| title_short | Variational methods for potential operator equations |
| title_sort | variational methods for potential operator equations with applications to nonlinear elliptic equations |
| title_sub | with applications to nonlinear elliptic equations |
| topic | Variationsproblem (DE-588)4187419-5 gnd Potenzialoperator (DE-588)4175486-4 gnd Elliptische Differentialgleichung (DE-588)4014485-9 gnd |
| topic_facet | Variationsproblem Potenzialoperator Elliptische Differentialgleichung |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=007570180&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
| volume_link | (DE-604)BV000005407 |
| work_keys_str_mv | AT chabrowskijan variationalmethodsforpotentialoperatorequationswithapplicationstononlinearellipticequations |