Implicit partial differential equations:
Saved in:
| Main Authors: | , |
|---|---|
| Format: | Book |
| Language: | English |
| Published: |
Basel [u.a.]
Birkhäuser
1999
|
| Series: | Progress in nonlinear differential equations and their applications
37 |
| Subjects: | |
| Online Access: | Inhaltsverzeichnis |
| Physical Description: | XII, 273 S. |
| ISBN: | 3764341211 0817641211 9780817641214 |
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MARC
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| 245 | 1 | 0 | |a Implicit partial differential equations |c Bernard Dacorogna ; Paolo Marcellini |
| 264 | 1 | |a Basel [u.a.] |b Birkhäuser |c 1999 | |
| 300 | |a XII, 273 S. | ||
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| 490 | 1 | |a Progress in nonlinear differential equations and their applications |v 37 | |
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Record in the Search Index
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|---|---|
| adam_text | Contents
Preface ix
Acknowledgments xi
1 Introduction 1
1.1 The first order case 1
1.1.1 Statement of the problem 1
1.1.2 The scalar case 2
1.1.3 Some examples in the vectorial case 4
1.1.4 Convexity conditions in the vectorial case 8
1.1.5 Some typical existence theorems in the vectorial case ... 9
1.2 Second and higher order cases 10
1.2.1 Dirichlet Neumann boundary value problem 10
1.2.2 Fully nonlinear partial differential equations 12
1.2.3 Singular values 13
1.2.4 Some extensions 14
1.3 Different methods 15
1.3.1 Viscosity solutions 15
1.3.2 Convex integration 17
1.3.3 The Baire category method 18
1.4 Applications to the calculus of variations 20
1.4.1 Some bibliographical notes 21
1.4.2 The variational problem 22
1.4.3 The scalar case 23
vi Contents
1.4.4 Application to optimal design in the vector valued case . . 24
1.5 Some unsolved problems 26
1.5.1 Selection criterion 26
1.5.2 Measurable Hamiltonians 26
1.5.3 Lipschitz boundary data 27
1.5.4 Approximation of Lipschitz functions by smooth functions 27
1.5.5 Extension of Lipschitz functions
and compatibility conditions 27
1.5.6 Existence under quasiconvexity assumption 28
1.5.7 Problems with constraints 28
1.5.8 Potential wells 29
1.5.9 Calculus of variations 30
1 First and Second Order PDE s 31
2 First Order Equations 33
2.1 Introduction 33
2.2 The convex case 34
2.2.1 The main theorem 34
2.2.2 An approximation lemma 36
2.2.3 The case independent of (x,u) 40
2.2.4 Proof of the main theorem 43
2.3 The nonconvex case 47
i 2.3.1 The pyramidal construction 47
2.3.2 The general case 52
2.4 The compatibility condition 56
2.5 An attainment result 60
3 Second Order Equations 69
3.1 Introduction 69
3.2 The convex case 70
3.2.1 Statement of the result and some examples 70
3.2.2 The approximation lemma 72
3.2.3 The case independent of lower order terms 73
3.2.4 Proof of the main theorem 77
3.3 Some extensions 81
3.3.1 Systems of convex functions 81
3.3.2 A problem with constraint on the determinant 82
3.3.3 Application to optimal design 90
4 Comparison with Viscosity Solutions 95
4.1 Introduction 95
4.2 Definition and examples 97
4.3 Geometric restrictions 100
Contents vii
4.3.1 Main results 100
4.3.2 Proof of the main results 103
4.4 Appendix 113
4.4.1 Subgradient and differentiability of convex functions ... 113
4.4.2 Gauges and their polars 113
4.4.3 Extension of Lipschitz functions 115
4.4.4 A property of the sub and super differentials 117
II Systems ofPartial Differential Equations 119
5 Some Preliminary Results 121
5.1 Introduction 121
5.2 Different notions of convexity 121
5.2.1 Definitions and basic properties (first order case) 121
5.2.2 Definitions and basic properties (higher order case) .... 124
5.2.3 Different envelopes 126
5.3 Weak lower semicontinuity 127
5.3.1 The first order case 127
5.3.2 The higher order case 129
5.4 Different notions of convexity for sets 130
5.4.1 Definitions 130
5.4.2 The different convex hulls 131
5.4.3 Further properties of rank one convex hulls 135
5.4.4 Extreme points 138
6 Existence Theorems for Systems 141
6.1 Introduction 141
6.2 An abstract result 142
6.2.1 The relaxation property 142
6.2.2 Weakly extreme sets 147
6.3 The key approximation lemma 148
6.4 Sufficient conditions for the relaxation property 152
6.4.1 One quasiconvex equation 152
6.4.2 The approximation property 153
6.4.3 Relaxation property for general sets 154
6.5 The main theorems 157
III Applications 167
7 The Singular Values Case 169
7.1 Introduction 169
7.2 Singular values and functions of singular values 171
7.2.1 Singular values 171
viii Contents
7.2.2 Functions depending on singular values 174
7.2.3 Rank one convexity in dimension two 181
7.3 Convex and rank one convex hulls 185
7.3.1 The case of equality of the a, 186
7.3.2 The main theorem for general matrices 187
7.3.3 The diagonal case in dimension two 193
7.3.4 The symmetric case in dimension two 195
7.4 Existence of solutions (the first order case) 199
7.5 Existence of solutions (the second order case) 200
8 The Case of Potential Wells 205
8.1 Introduction 205
8.2 The rank one convex hull 206
8.3 Existence of solutions 215
9 The Complex Eikonal Equation 217
9.1 Introduction 217
9.2 The convex and rank one convex hulls 218
9.3 Existence of solutions 222
IV Appendix 223
10 Appendix: Piecewise Approximations 225
10.1 Vitali covering theorems and applications 225
10.1.1 Vitali covering theorems 225
10.1.2 Piecewise affine approximation 232
10.2 Piecewise polynomial approximation 241
10.2.1 Approximation of functions of class CN 242
10.2.2 Approximation of functions of class WN °° 245
References 249
Index 271
|
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| illustrated | Not Illustrated |
| indexdate | 2024-07-09T18:33:21Z |
| institution | BVB |
| isbn | 3764341211 0817641211 9780817641214 |
| language | English |
| oai_aleph_id | oai:aleph.bib-bvb.de:BVB01-008683678 |
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| physical | XII, 273 S. |
| publishDate | 1999 |
| publishDateSearch | 1999 |
| publishDateSort | 1999 |
| publisher | Birkhäuser |
| record_format | marc |
| series | Progress in nonlinear differential equations and their applications |
| series2 | Progress in nonlinear differential equations and their applications |
| spelling | Dacorogna, Bernard 1953- Verfasser (DE-588)133791920 aut Implicit partial differential equations Bernard Dacorogna ; Paolo Marcellini Basel [u.a.] Birkhäuser 1999 XII, 273 S. txt rdacontent n rdamedia nc rdacarrier Progress in nonlinear differential equations and their applications 37 Nichtlineare partielle Differentialgleichung (DE-588)4128900-6 gnd rswk-swf Nichtlineare partielle Differentialgleichung (DE-588)4128900-6 s DE-604 Marcellini, Paolo Verfasser aut Progress in nonlinear differential equations and their applications 37 (DE-604)BV007934389 37 HBZ Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=008683678&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Dacorogna, Bernard 1953- Marcellini, Paolo Implicit partial differential equations Progress in nonlinear differential equations and their applications Nichtlineare partielle Differentialgleichung (DE-588)4128900-6 gnd |
| subject_GND | (DE-588)4128900-6 |
| title | Implicit partial differential equations |
| title_auth | Implicit partial differential equations |
| title_exact_search | Implicit partial differential equations |
| title_full | Implicit partial differential equations Bernard Dacorogna ; Paolo Marcellini |
| title_fullStr | Implicit partial differential equations Bernard Dacorogna ; Paolo Marcellini |
| title_full_unstemmed | Implicit partial differential equations Bernard Dacorogna ; Paolo Marcellini |
| title_short | Implicit partial differential equations |
| title_sort | implicit partial differential equations |
| topic | Nichtlineare partielle Differentialgleichung (DE-588)4128900-6 gnd |
| topic_facet | Nichtlineare partielle Differentialgleichung |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=008683678&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
| volume_link | (DE-604)BV007934389 |
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