A geometric approach to free boundary problems:
Gespeichert in:
| Hauptverfasser: | , |
|---|---|
| Format: | Buch |
| Sprache: | English |
| Veröffentlicht: |
Providence, Rhode Island
American Mathematical Society
[2005]
|
| Schriftenreihe: | Graduate studies in mathematics
68 |
| Schlagworte: | |
| Online-Zugang: | Inhaltsverzeichnis |
| Beschreibung: | Hier auch später erschienene, unveränderte Nachdrucke |
| Beschreibung: | ix, 270 Seiten Illustrationen 26 cm |
| ISBN: | 0821837842 9780821837849 |
Internformat
MARC
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| 100 | 1 | |a Caffarelli, Luis |d 1948- |e Verfasser |0 (DE-588)1081049367 |4 aut | |
| 245 | 1 | 0 | |a A geometric approach to free boundary problems |c Luis Caffarelli ; Sandro Salsa |
| 264 | 1 | |a Providence, Rhode Island |b American Mathematical Society |c [2005] | |
| 300 | |a ix, 270 Seiten |b Illustrationen |c 26 cm | ||
| 336 | |b txt |2 rdacontent | ||
| 337 | |b n |2 rdamedia | ||
| 338 | |b nc |2 rdacarrier | ||
| 490 | 1 | |a Graduate studies in mathematics |v 68 | |
| 500 | |a Hier auch später erschienene, unveränderte Nachdrucke | ||
| 650 | 7 | |a Equações diferenciais parciais |2 larpcal | |
| 650 | 4 | |a Lipschitz, Espaces de | |
| 650 | 7 | |a Problemas de contorno |2 larpcal | |
| 650 | 4 | |a Problèmes aux limites | |
| 650 | 4 | |a Boundary value problems | |
| 650 | 4 | |a Lipschitz spaces | |
| 650 | 0 | 7 | |a Freies Randwertproblem |0 (DE-588)4155303-2 |2 gnd |9 rswk-swf |
| 650 | 0 | 7 | |a Lipschitz-Raum |0 (DE-588)4431681-1 |2 gnd |9 rswk-swf |
| 689 | 0 | 0 | |a Freies Randwertproblem |0 (DE-588)4155303-2 |D s |
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Datensatz im Suchindex
| _version_ | 1804134549632843776 |
|---|---|
| adam_text | Contents
Introduction vii
Part 1. Elliptic Problems
Chapter 1. An Introductory Problem 3
§1.1. Introduction and heuristic considerations 3
§1.2. A one phase singular perturbation problem 6
§1.3. The free boundary condition 17
Chapter 2. Viscosity Solutions and Their Asymptotic Developments 25
§2.1. The notion of viscosity solution 25
§2.2. Asymptotic developments 27
§2.3. Comparison principles 30
Chapter 3. The Regularity of the Free Boundary 35
§3.1. Weak results 35
§3.2. Weak results for one phase problems 36
§3.3. Strong results 40
Chapter 4. Lipschitz Free Boundaries Are C1 7 43
§4.1. The main theorem. Heuristic considerations and strategy 43
§4.2. Interior improvement of the Lipschitz constant 47
§4.3. A Harnack principle. Improved interior gain 51
§4.4. A continuous family of .R subsolutions 53
§4.5. Free boundary improvement. Basic iteration 62
iii
iv Contents
Chapter 5. Flat Free Boundaries Are Lipschitz 65
§5.1. Heuristic considerations 65
§5.2. An auxiliary family of functions 70
§5.3. Level surfaces of normal perturbations of e monotone
functions 72
§5.4. A continuous family of _R subsolutions 74
§5.5. Proof of Theorem 5.1 76
§5.6. A degenerate case 80
Chapter 6. Existence Theory 87
§6.1. Introduction 87
§6.2. u+ is locally Lipschitz 90
§6.3. u is Lipschitz 91
§6.4. u+ is nondegenerate 95
§6.5. u is a viscosity supersolution 96
§6.6. it is a viscosity subsolution 99
§6.7. Measure theoretic properties of F(u) 101
§6.8. Asymptotic developments 103
§6.9. Regularity and compactness 106
Part 2. Evolution Problems
Chapter 7. Parabolic Free Boundary Problems 111
§7.1. Introduction 111
§7.2. A class of free boundary problems and their viscosity
solutions 113
§7.3. Asymptotic behavior and free boundary relation 116
§7.4. i? subsolutions and a comparison principle 118
Chapter 8. Lipschitz Free Boundaries: Weak Results 121
§8.1. Lipschitz continuity of viscosity solutions 121
§8.2. Asymptotic behavior and free boundary relation 124
§8.3. Counterexamples 125
Chapter 9. Lipschitz Free Boundaries: Strong Results 131
§9.1. Nondegenerate problems: main result and strategy 131
§9.2. Interior gain in space (parabolic homogeneity) 135
§9.3. Common gain 138
§9.4. Interior gain in space (hyperbolic homogeneity) 141
Contents v
§9.5. Interior gain in time 143
§9.6. A continuous family of subcaloric functions 149
§9.7. Free boundary improvement. Propagation lemma 153
§9.8. Regularization of the free boundary in space 157
§9.9. Free boundary regularity in space and time 160
Chapter 10. Flat Free Boundaries Are Smooth 165
§10.1. Main result and strategy 165
§10.2. Interior enlargement of the monotonicity cone 168
§10.3. Control of uv at a contact point 172
§10.4. A continuous family of perturbations 174
§10.5. Improvement of e monotonicity 177
§10.6. Propagation of cone enlargement to the free boundary 180
§10.7. Proof of the main theorem 183
§10.8. Finite time regularization 185
Part 3. Complementary Chapters: Main Tools
Chapter 11. Boundary Behavior of Harmonic Functions 191
§11.1. Harmonic functions in Lipschitz domains 191
§11.2. Boundary Harnack principles 195
§11.3. An excursion on harmonic measure 201
§11.4. Monotonicity properties 203
§11.5. e monotonicity and full monotonicity 205
§11.6. Linear behavior at regular boundary points 207
Chapter 12. Monotonicity Formulas and Applications 211
§12.1. A 2 dimensional formula 211
§12.2. The n dimensional formula 214
§12.3. Consequences and applications 222
§12.4. A parabolic monotonicity formula 230
§12.5. A singular perturbation parabolic problem 233
Chapter 13. Boundary Behavior of Caloric Functions 235
§13.1. Caloric functions in Lip(l, 1/2) domains 235
§13.2. Caloric functions in Lipschitz domains 241
§13.3. Asymptotic behavior near the zero set 248
§13.4. e monotonicity and full monotonicity 256
vi Contents
§13.5. An excursion on caloric measure 262
Bibliography 265
Index 269
|
| adam_txt |
Contents
Introduction vii
Part 1. Elliptic Problems
Chapter 1. An Introductory Problem 3
§1.1. Introduction and heuristic considerations 3
§1.2. A one phase singular perturbation problem 6
§1.3. The free boundary condition 17
Chapter 2. Viscosity Solutions and Their Asymptotic Developments 25
§2.1. The notion of viscosity solution 25
§2.2. Asymptotic developments 27
§2.3. Comparison principles 30
Chapter 3. The Regularity of the Free Boundary 35
§3.1. Weak results 35
§3.2. Weak results for one phase problems 36
§3.3. Strong results 40
Chapter 4. Lipschitz Free Boundaries Are C1'7 43
§4.1. The main theorem. Heuristic considerations and strategy 43
§4.2. Interior improvement of the Lipschitz constant 47
§4.3. A Harnack principle. Improved interior gain 51
§4.4. A continuous family of .R subsolutions 53
§4.5. Free boundary improvement. Basic iteration 62
iii
iv Contents
Chapter 5. Flat Free Boundaries Are Lipschitz 65
§5.1. Heuristic considerations 65
§5.2. An auxiliary family of functions 70
§5.3. Level surfaces of normal perturbations of e monotone
functions 72
§5.4. A continuous family of _R subsolutions 74
§5.5. Proof of Theorem 5.1 76
§5.6. A degenerate case 80
Chapter 6. Existence Theory 87
§6.1. Introduction 87
§6.2. u+ is locally Lipschitz 90
§6.3. u is Lipschitz 91
§6.4. u+ is nondegenerate 95
§6.5. u is a viscosity supersolution 96
§6.6. it is a viscosity subsolution 99
§6.7. Measure theoretic properties of F(u) 101
§6.8. Asymptotic developments 103
§6.9. Regularity and compactness 106
Part 2. Evolution Problems
Chapter 7. Parabolic Free Boundary Problems 111
§7.1. Introduction 111
§7.2. A class of free boundary problems and their viscosity
solutions 113
§7.3. Asymptotic behavior and free boundary relation 116
§7.4. i? subsolutions and a comparison principle 118
Chapter 8. Lipschitz Free Boundaries: Weak Results 121
§8.1. Lipschitz continuity of viscosity solutions 121
§8.2. Asymptotic behavior and free boundary relation 124
§8.3. Counterexamples 125
Chapter 9. Lipschitz Free Boundaries: Strong Results 131
§9.1. Nondegenerate problems: main result and strategy 131
§9.2. Interior gain in space (parabolic homogeneity) 135
§9.3. Common gain 138
§9.4. Interior gain in space (hyperbolic homogeneity) 141
Contents v
§9.5. Interior gain in time 143
§9.6. A continuous family of subcaloric functions 149
§9.7. Free boundary improvement. Propagation lemma 153
§9.8. Regularization of the free boundary in space 157
§9.9. Free boundary regularity in space and time 160
Chapter 10. Flat Free Boundaries Are Smooth 165
§10.1. Main result and strategy 165
§10.2. Interior enlargement of the monotonicity cone 168
§10.3. Control of uv at a "contact point" 172
§10.4. A continuous family of perturbations 174
§10.5. Improvement of e monotonicity 177
§10.6. Propagation of cone enlargement to the free boundary 180
§10.7. Proof of the main theorem 183
§10.8. Finite time regularization 185
Part 3. Complementary Chapters: Main Tools
Chapter 11. Boundary Behavior of Harmonic Functions 191
§11.1. Harmonic functions in Lipschitz domains 191
§11.2. Boundary Harnack principles 195
§11.3. An excursion on harmonic measure 201
§11.4. Monotonicity properties 203
§11.5. e monotonicity and full monotonicity 205
§11.6. Linear behavior at regular boundary points 207
Chapter 12. Monotonicity Formulas and Applications 211
§12.1. A 2 dimensional formula 211
§12.2. The n dimensional formula 214
§12.3. Consequences and applications 222
§12.4. A parabolic monotonicity formula 230
§12.5. A singular perturbation parabolic problem 233
Chapter 13. Boundary Behavior of Caloric Functions 235
§13.1. Caloric functions in Lip(l, 1/2) domains 235
§13.2. Caloric functions in Lipschitz domains 241
§13.3. Asymptotic behavior near the zero set 248
§13.4. e monotonicity and full monotonicity 256
vi Contents
§13.5. An excursion on caloric measure 262
Bibliography 265
Index 269 |
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| illustrated | Illustrated |
| index_date | 2024-07-02T13:16:49Z |
| indexdate | 2024-07-09T20:26:23Z |
| institution | BVB |
| isbn | 0821837842 9780821837849 |
| language | English |
| lccn | 2005041181 |
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| spelling | Caffarelli, Luis 1948- Verfasser (DE-588)1081049367 aut A geometric approach to free boundary problems Luis Caffarelli ; Sandro Salsa Providence, Rhode Island American Mathematical Society [2005] ix, 270 Seiten Illustrationen 26 cm txt rdacontent n rdamedia nc rdacarrier Graduate studies in mathematics 68 Hier auch später erschienene, unveränderte Nachdrucke Equações diferenciais parciais larpcal Lipschitz, Espaces de Problemas de contorno larpcal Problèmes aux limites Boundary value problems Lipschitz spaces Freies Randwertproblem (DE-588)4155303-2 gnd rswk-swf Lipschitz-Raum (DE-588)4431681-1 gnd rswk-swf Freies Randwertproblem (DE-588)4155303-2 s Lipschitz-Raum (DE-588)4431681-1 s 1\p DE-604 Salsa, Sandro 1950- Verfasser (DE-588)1111815445 aut Erscheint auch als Online-Ausgabe 978-1-4704-2109-0 Graduate studies in mathematics 68 (DE-604)BV009739289 68 HBZ Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=014163566&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis 1\p cgwrk 20201028 DE-101 https://d-nb.info/provenance/plan#cgwrk |
| spellingShingle | Caffarelli, Luis 1948- Salsa, Sandro 1950- A geometric approach to free boundary problems Graduate studies in mathematics Equações diferenciais parciais larpcal Lipschitz, Espaces de Problemas de contorno larpcal Problèmes aux limites Boundary value problems Lipschitz spaces Freies Randwertproblem (DE-588)4155303-2 gnd Lipschitz-Raum (DE-588)4431681-1 gnd |
| subject_GND | (DE-588)4155303-2 (DE-588)4431681-1 |
| title | A geometric approach to free boundary problems |
| title_auth | A geometric approach to free boundary problems |
| title_exact_search | A geometric approach to free boundary problems |
| title_exact_search_txtP | A geometric approach to free boundary problems |
| title_full | A geometric approach to free boundary problems Luis Caffarelli ; Sandro Salsa |
| title_fullStr | A geometric approach to free boundary problems Luis Caffarelli ; Sandro Salsa |
| title_full_unstemmed | A geometric approach to free boundary problems Luis Caffarelli ; Sandro Salsa |
| title_short | A geometric approach to free boundary problems |
| title_sort | a geometric approach to free boundary problems |
| topic | Equações diferenciais parciais larpcal Lipschitz, Espaces de Problemas de contorno larpcal Problèmes aux limites Boundary value problems Lipschitz spaces Freies Randwertproblem (DE-588)4155303-2 gnd Lipschitz-Raum (DE-588)4431681-1 gnd |
| topic_facet | Equações diferenciais parciais Lipschitz, Espaces de Problemas de contorno Problèmes aux limites Boundary value problems Lipschitz spaces Freies Randwertproblem Lipschitz-Raum |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=014163566&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
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