Convex variational problems: linear, nearly linear, and anisotropic growth conditions
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| Main Author: | |
|---|---|
| Format: | Book |
| Language: | English |
| Published: |
Berlin
Springer
2003
|
| Series: | Lecture notes in mathematics
1818 |
| Subjects: | |
| Online Access: | Inhaltsverzeichnis |
| Physical Description: | X, 217 S. graph. Darst. |
| ISBN: | 3540402985 |
Staff View
MARC
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| 100 | 1 | |a Bildhauer, Michael |e Verfasser |4 aut | |
| 245 | 1 | 0 | |a Convex variational problems |b linear, nearly linear, and anisotropic growth conditions |c Michael Bildhauer |
| 264 | 1 | |a Berlin |b Springer |c 2003 | |
| 300 | |a X, 217 S. |b graph. Darst. | ||
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| 650 | 4 | |a Variationsproblem | |
| 650 | 4 | |a Calculus of variations | |
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Record in the Search Index
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| adam_text | Contents
1 Introduction 1
2 Variational problems with linear growth: the general setting 13
2.1 Construction of a solution for the dual problem which is of
class Wl^SlW) 14
2.1.1 The dual problem 14
2.1.2 Regularization 16
2.1.3 W^^ regularity for the dual problem 19
2.2 A uniqueness theorem for the dual problem 20
2.3 Partial C1 01 and C° a regularity, respectively, for generalized
minimizers and for the dual solution 25
2.3.1 Partial ^ regularity of generalized minimizers 26
2.3.2 Partial C° Q regularity of the dual solution 29
2.4 Degenerate variational problems with linear growth 32
2.4.1 The duality relation for degenerate problems 33
2.4.2 Application: an intrinsic regularity theory for a 39
3 Variational integrands with (s, fj,, q) growth 41
3.1 Existence in Orlicz Sobolev spaces 42
3.2 The notion of (s, fi, g) growth examples 44
3.3 A priori gradient bounds and local C1 estimates for scalar
and structured vector valued problems 50
3.3.1 Regularization 52
3.3.2 A priori L9 estimates 54
3.3.3 Proof of Theorem 3.16 61
3.3.4 Conclusion 67
3.4 Partial regularity in the general vectorial setting 69
3.4.1 Regularization 69
3.4.2 A Caccioppoli type inequality 70
3.4.3 Blow up 72
3.4.3.1 Blow up and limit equation 74
3.4.3.2 An auxiliary proposition 76
3.4.3.3 Strong convergence 83
3.4.3.4 Conclusion 86
3.4.4 Iteration 87
X Contents
3.5 Comparison with some known results 89
3.5.1 The scalar case 89
3.5.2 The vectorial setting 90
3.6 Two dimensional anisotropic variational problems 91
4 Variational problems with linear growth: the case of
^ elliptic integrands 97
4.1 The case /x 1 + 2/n 100
4.1.1 Regularization 101
4.1.2 Some remarks on the dual problem 101
4.1.3 Proof of Theorem 4.4 103
4.2 Bounded generalized solutions 104
4.2.1 Regularization 108
4.2.2 The limit case /j, = 3 111
4.2.2.1 Higher local integrability 111
4.2.2.2 The independent variable 113
4.2.3 Lp estimates in the case fj, 3 116
4.2.4 A priori gradient bounds 118
4.3 Two dimensional problems 122
4.3.1 Higher local integrability in the limit case 123
4.3.2 The case fi 3 129
4.4 A counterexample 132
5 Bounded solutions for convex variational problems with a
wide range of anisotropy 141
5.1 Vector valued problems 142
5.2 Scalar obstacle problems 149
6 Anisotropic linear/superlinear growth in the scalar case ... 161
A Some remarks on relaxation 173
A.I The approach known from the minimal surface case 174
A.2 The approach known from the theory of perfect plasticity 176
A.3 Two uniqueness results 181
B Some density results 185
B.I Approximations in BV 185
B.2 A density result for U n L(c) 191
B.3 Local comparison functions 194
C Brief comments on steady states of generalized Newtonian
fluids 199
D Notation and conventions 205
References 207
Index 215
|
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| indexdate | 2024-07-09T19:15:44Z |
| institution | BVB |
| isbn | 3540402985 |
| language | English |
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| physical | X, 217 S. graph. Darst. |
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| series2 | Lecture notes in mathematics |
| spelling | Bildhauer, Michael Verfasser aut Convex variational problems linear, nearly linear, and anisotropic growth conditions Michael Bildhauer Berlin Springer 2003 X, 217 S. graph. Darst. txt rdacontent n rdamedia nc rdacarrier Lecture notes in mathematics 1818 Variationsproblem Calculus of variations Differential equations, Elliptic Numerical solutions Variationsproblem (DE-588)4187419-5 gnd rswk-swf Variationsproblem (DE-588)4187419-5 s DE-604 Lecture notes in mathematics 1818 (DE-604)BV000676446 1818 HBZ Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=010399350&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Bildhauer, Michael Convex variational problems linear, nearly linear, and anisotropic growth conditions Lecture notes in mathematics Variationsproblem Calculus of variations Differential equations, Elliptic Numerical solutions Variationsproblem (DE-588)4187419-5 gnd |
| subject_GND | (DE-588)4187419-5 |
| title | Convex variational problems linear, nearly linear, and anisotropic growth conditions |
| title_auth | Convex variational problems linear, nearly linear, and anisotropic growth conditions |
| title_exact_search | Convex variational problems linear, nearly linear, and anisotropic growth conditions |
| title_full | Convex variational problems linear, nearly linear, and anisotropic growth conditions Michael Bildhauer |
| title_fullStr | Convex variational problems linear, nearly linear, and anisotropic growth conditions Michael Bildhauer |
| title_full_unstemmed | Convex variational problems linear, nearly linear, and anisotropic growth conditions Michael Bildhauer |
| title_short | Convex variational problems |
| title_sort | convex variational problems linear nearly linear and anisotropic growth conditions |
| title_sub | linear, nearly linear, and anisotropic growth conditions |
| topic | Variationsproblem Calculus of variations Differential equations, Elliptic Numerical solutions Variationsproblem (DE-588)4187419-5 gnd |
| topic_facet | Variationsproblem Calculus of variations Differential equations, Elliptic Numerical solutions |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=010399350&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
| volume_link | (DE-604)BV000676446 |
| work_keys_str_mv | AT bildhauermichael convexvariationalproblemslinearnearlylinearandanisotropicgrowthconditions |