The maximum principle:
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| Main Authors: | , |
|---|---|
| Format: | Book |
| Language: | English |
| Published: |
Basel [u.a.]
Birkhäuser
2007
|
| Series: | Progress in non-linear differential equations and their applications
73 |
| Subjects: | |
| Online Access: | Inhaltsverzeichnis |
| Item Description: | Literaturverz. S. 223 - 232 Auch als Internetausgabe |
| Physical Description: | X, 234 S. graph. Darst. |
| ISBN: | 9783764381448 3764381442 |
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MARC
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| 245 | 1 | 0 | |a The maximum principle |c Patrizia Pucci ; James Serrin |
| 264 | 1 | |a Basel [u.a.] |b Birkhäuser |c 2007 | |
| 300 | |a X, 234 S. |b graph. Darst. | ||
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Record in the Search Index
| _version_ | 1804137126719127552 |
|---|---|
| adam_text | Contents
Preface
.................................. ix
1
Introduction
and Preliminaries
1.1
Introduction
........................... 1
1.2
Notation
............................. 10
2
Tangency and Comparison Theorems for Elliptic Inequalities
2.1
The contributions of
Eberhard Hopf............. 13
2.2
Tangency and comparison principles for
quasilinear
inequalities
........................... 21
2.3
Maximum and sweeping principles for
quasilinear
inequalities
........................... 25
2.4
Comparison theorems for divergence structure
inequalities
........................... 30
2.5
Tangency theorems via Harnack s inequality
......... 34
2.6
Uniqueness of the Dirichlet problem
............. 37
2.7
The boundary point lemma
.................. 39
2.8
Appendix: Proof of
Eberhard Hopfs
maximum principle
....................... 42
Notes
................................. 46
Problems
............................... 46
3
Maximum Principles for Divergence Structure Elliptic
Differential Inequalities
3.1
Distribution solutions
..................... 51
3.2
Maximum principles for homogeneous inequalities
...... 54
3.3
A maximum principle for thin sets
.............. 59
vi
Contents
3.4
A
comparison theorem in
W1>p(íž)..............
61
3.5
Comparison theorems for singular elliptic inequalities
.... 62
3.6
Strongly degenerate operators
................. 68
3.7
Maximum principles for non-homogeneous elliptic
inequalities
........................... 72
3.8
Uniqueness of the singular Dirichlet problem
........ 78
3.9
Appendix: Sobolev s inequality
................ 79
Notes
................................. 81
Problems
............................... 81
4
Boundary Value Problems for Nonlinear Ordinary
Differential Equations
4.1
Preliminary lemmas
...................... 83
4.2
Existence theorems
....................... 89
4.3
Existence theorems on a half-line
............... 92
4.4
The end point lemma
..................... 96
4.5
Appendix: Proof of Proposition
4.2.1............. 97
Problems
............................... 101
5
The Strong Maximum Principle and the Compact
Support Principle
5.1
The strong maximum principle
................ 103
5.2
The compact support principle
................ 105
5.3
A special case
.......................... 107
5.4
Strong maximum principle: Generalized version
....... 110
5.5
A boundary point lemma
................... 119
5.6
Compact support principle: Generalized version
....... 120
Notes
................................. 125
Problems
............................... 126
6
Non-homogeneous Divergence Structure Inequalities
6.1
Maximum principles for structured inequalities
....... 127
6.2
Proof of Theorems
6.1.1
and
6.1.2 .............. 131
6.3
Proof of Theorem
6.1.3
and the first part
of Theorem
6.1.5........................ 139
6.4
Proof of Theorem
6.1.4
and the second part
of Theorem
6.1.5........................ 142
Contents
vii
6.5
The case
ρ =
1
and the mean curvature equation
...... 146
Notes
................................. 150
Problems
............................... 150
7
The Harnack Inequality
7.1
Local boundedness and the weak Harnack inequality
.... 153
7.2
The Harnack inequality
.................... 163
7.3
Holder continuity
........................ 166
7.4
The case
ρ > η.........................
171
7.5
Appendix. The John-Nirenberg theorem
........... 173
Notes
................................. 179
Problems
............................... 180
8
Applications
8.1
Cauchy-Liouville Theorems
.................. 181
8.2
Radial symmetry
........................ 186
8.3
Symmetry for over determined boundary value
problems
............................. 195
8.4
The phenomenon of dead cores
................ 203
8.5
The strong maximum principle for Riemannian
manifolds
............................ 218
Problems
............................... 220
Bibliography
............................... 223
Subject Index
............................... 233
Author Index
............................... 235
|
| adam_txt |
Contents
Preface
. ix
1
Introduction
and Preliminaries
1.1
Introduction
. 1
1.2
Notation
. 10
2
Tangency and Comparison Theorems for Elliptic Inequalities
2.1
The contributions of
Eberhard Hopf. 13
2.2
Tangency and comparison principles for
quasilinear
inequalities
. 21
2.3
Maximum and sweeping principles for
quasilinear
inequalities
. 25
2.4
Comparison theorems for divergence structure
inequalities
. 30
2.5
Tangency theorems via Harnack's inequality
. 34
2.6
Uniqueness of the Dirichlet problem
. 37
2.7
The boundary point lemma
. 39
2.8
Appendix: Proof of
Eberhard Hopfs
maximum principle
. 42
Notes
. 46
Problems
. 46
3
Maximum Principles for Divergence Structure Elliptic
Differential Inequalities
3.1
Distribution solutions
. 51
3.2
Maximum principles for homogeneous inequalities
. 54
3.3
A maximum principle for thin sets
. 59
vi
Contents
3.4
A
comparison theorem in
W1>p(íž).
61
3.5
Comparison theorems for singular elliptic inequalities
. 62
3.6
Strongly degenerate operators
. 68
3.7
Maximum principles for non-homogeneous elliptic
inequalities
. 72
3.8
Uniqueness of the singular Dirichlet problem
. 78
3.9
Appendix: Sobolev's inequality
. 79
Notes
. 81
Problems
. 81
4
Boundary Value Problems for Nonlinear Ordinary
Differential Equations
4.1
Preliminary lemmas
. 83
4.2
Existence theorems
. 89
4.3
Existence theorems on a half-line
. 92
4.4
The end point lemma
. 96
4.5
Appendix: Proof of Proposition
4.2.1. 97
Problems
. 101
5
The Strong Maximum Principle and the Compact
Support Principle
5.1
The strong maximum principle
. 103
5.2
The compact support principle
. 105
5.3
A special case
. 107
5.4
Strong maximum principle: Generalized version
. 110
5.5
A boundary point lemma
. 119
5.6
Compact support principle: Generalized version
. 120
Notes
. 125
Problems
. 126
6
Non-homogeneous Divergence Structure Inequalities
6.1
Maximum principles for structured inequalities
. 127
6.2
Proof of Theorems
6.1.1
and
6.1.2 . 131
6.3
Proof of Theorem
6.1.3
and the first part
of Theorem
6.1.5. 139
6.4
Proof of Theorem
6.1.4
and the second part
of Theorem
6.1.5. 142
Contents
vii
6.5
The case
ρ =
1
and the mean curvature equation
. 146
Notes
. 150
Problems
. 150
7
The Harnack Inequality
7.1
Local boundedness and the weak Harnack inequality
. 153
7.2
The Harnack inequality
. 163
7.3
Holder continuity
. 166
7.4
The case
ρ > η.
171
7.5
Appendix. The John-Nirenberg theorem
. 173
Notes
. 179
Problems
. 180
8
Applications
8.1
Cauchy-Liouville Theorems
. 181
8.2
Radial symmetry
. 186
8.3
Symmetry for over determined boundary value
problems
. 195
8.4
The phenomenon of dead cores
. 203
8.5
The strong maximum principle for Riemannian
manifolds
. 218
Problems
. 220
Bibliography
. 223
Subject Index
. 233
Author Index
. 235 |
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| illustrated | Illustrated |
| index_date | 2024-07-02T18:46:25Z |
| indexdate | 2024-07-09T21:07:20Z |
| institution | BVB |
| isbn | 9783764381448 3764381442 |
| language | English |
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| record_format | marc |
| series | Progress in non-linear differential equations and their applications |
| series2 | Progress in non-linear differential equations and their applications |
| spelling | Pucci, Patrizia 1952- Verfasser (DE-588)133429261 aut The maximum principle Patrizia Pucci ; James Serrin Basel [u.a.] Birkhäuser 2007 X, 234 S. graph. Darst. txt rdacontent n rdamedia nc rdacarrier Progress in non-linear differential equations and their applications 73 Literaturverz. S. 223 - 232 Auch als Internetausgabe Differential equations, Partial Maximum principles (Mathematics) Maximumprinzip (DE-588)4169165-9 gnd rswk-swf Elliptische Differentialgleichung (DE-588)4014485-9 gnd rswk-swf Elliptische Differentialgleichung (DE-588)4014485-9 s Maximumprinzip (DE-588)4169165-9 s DE-604 Serrin, James 1926-2012 Verfasser (DE-588)118892924 aut Progress in non-linear differential equations and their applications 73 (DE-604)BV007934389 73 Digitalisierung UB Regensburg application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=016074020&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Pucci, Patrizia 1952- Serrin, James 1926-2012 The maximum principle Progress in non-linear differential equations and their applications Differential equations, Partial Maximum principles (Mathematics) Maximumprinzip (DE-588)4169165-9 gnd Elliptische Differentialgleichung (DE-588)4014485-9 gnd |
| subject_GND | (DE-588)4169165-9 (DE-588)4014485-9 |
| title | The maximum principle |
| title_auth | The maximum principle |
| title_exact_search | The maximum principle |
| title_exact_search_txtP | The maximum principle |
| title_full | The maximum principle Patrizia Pucci ; James Serrin |
| title_fullStr | The maximum principle Patrizia Pucci ; James Serrin |
| title_full_unstemmed | The maximum principle Patrizia Pucci ; James Serrin |
| title_short | The maximum principle |
| title_sort | the maximum principle |
| topic | Differential equations, Partial Maximum principles (Mathematics) Maximumprinzip (DE-588)4169165-9 gnd Elliptische Differentialgleichung (DE-588)4014485-9 gnd |
| topic_facet | Differential equations, Partial Maximum principles (Mathematics) Maximumprinzip Elliptische Differentialgleichung |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=016074020&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
| volume_link | (DE-604)BV007934389 |
| work_keys_str_mv | AT puccipatrizia themaximumprinciple AT serrinjames themaximumprinciple |