An introduction to gamma-convergence:
Saved in:
| Main Author: | |
|---|---|
| Format: | Book |
| Language: | English |
| Published: |
Boston [u.a.]
Birkhäuser
[1993]
|
| Series: | Progress in nonlinear differential equations and their applications
Volume 8 |
| Subjects: | |
| Online Access: | Inhaltsverzeichnis |
| Item Description: | Literaturverz. Seite [289] - 337 |
| Physical Description: | xiv, 340 Seiten |
| ISBN: | 081763679X 376433679X 9780817636791 |
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| 245 | 1 | 0 | |a An introduction to gamma-convergence |c Gianni Dal Maso |
| 264 | 1 | |a Boston [u.a.] |b Birkhäuser |c [1993] | |
| 264 | 4 | |c © 1993 | |
| 300 | |a xiv, 340 Seiten | ||
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Record in the Search Index
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| adam_text |
Contents
Introduction 1
1. The direct method in the calculus of variations 8
Lower semicontinuity
Sequential lower semicontinuity
Epigraphs
Coerciveness
Minimizing sequences
Existence of minimum points
Convexity and strict convexity
Uniqueness of minimum points
Example of lower semicontinuous integral functional in Lp spaces
The Caratheodory Continuity Theorem
The Sobolev spaces W1* and W^'"
Example of lower semicontinuous integral functionals in Sobolev spaces
2. Minimum problems for integral functionals 19
Existence of minimum points for integral functionals in Sobolev spaces
The case of Dirichlet boundary conditions
The case of weakly closed constraints
Comparison between strong V and weak Wl'p lower semicontinuity
Comparison between strong Lp and weak W^'v coerciveness
3. Relaxation 28
Lower semicontinuous envelopes
Sequential characterization of the lower semicontinuous envelope
Connection between the original problem and the relaxed problem
The case of convex functions
Example of relaxation of integral functionals
The Sobolev spaces obtained by a relaxation argument
The space BV obtained by a relaxation argument
4. F convergence and K convergence 38
T lower limit, F upper limit, F limit
Local character of F limits
Simple examples of F limits of functions of one real variable
F limit of a constant sequence and relaxation
The F lower limit as a lower limit in a suitable product space
Continuous convergence
K lower limit, K upper limit, K limit
The K lower limit obtained from simple closure operations
K limits expressed in terms of F limits of the indicator functions
F limits expressed in terms of K limits of the epigraphs
F limits expressed in terms of K limits of the level sets
viii An Introduction to F convergence
5. Comparison with pointwise convergence 46
Inequalities between F limits and ordinary limits
F convergence and uniform convergence
T limits of increasing sequences
T limits of decreasing sequences
F limits of equi lower semicontinuous sequences
Convexity and equi continuity
F limits of sequences of locally equi bounded convex functions
Convergence of integrands and F convergence of functional
6. Some properties of F limits 53
F limits and K limits of subsequences
F limits and K limits with respect to different topologies
Sequences with different F limits in the strong and in the weak topology
Monotonicity of F limits and K limits
Semicontinuity of F limits and closedness of K limits
F limits of relaxed functionals and K limits of closures
F limits of restrictions of functions
F limits of superpositions
F limits of the sum of two sequences
F limits of truncations
7. Convergence of minima and of minimizers 67
Inequalities on compact and open sets
Convergence of minima
Equi coercive sequences
Convergence of minima for equi coercive sequences
£ minimizers
Inclusion properties of the e minimizers of the F limits
Limits of e minimizers and minimizers of the F limit
The equi coercive case
8. Sequential characterization of F limits 86
F limits and K limits in spaces satisfying the first axiom of countability
Urysohn property of F limits and K limits
Compactness of F limits and K limits in spaces with a countable base
Metrizability of the weak topology in a Banach space with a separable dual
F limits and K limits in the weak topology of a Banach space
The case of a separable dual
The reflexive case
Urysohn property of F limits in the weak topology of a Banach space
9. F convergence in metric spaces 101
Characterization of F limits in completely regular spaces
Equivalence between F convergence and convergence of minima
Characterization of F limits in completely regular spaces
Moreau Yosida approximation of arbitrary functions in metric spaces
Connections with Holder continuous functions
Equi continuity of the Moreau Yosida approximations
F convergence and convergence of the Moreau Yosida approximations
Contents ix
10. The topology of T convergence Ill
Three topologies on the space of all lower semicontinuous functions
Connection of these topologies with the problem of convergence of minima
Compactness properties of these topologies
Relationships between these topologies and F convergence
k spaces
Topologizability of F convergence in the case of k spaces
Lack of separation properties of the topology of F convergence
Metrizability of F convergence for equi coercive functions on metric spaces
11. F convergence in topological vector spaces 126
F limits of convex functions
F limits of even functions
F limits of positively homogeneous functions
Quadratic forms with extended real values
Characterization of quadratic forms in terms of the parallelogram identity
F limits of quadratic forms
F limits of Holder continuous functions
12. Quadratic forms and linear operators 133
Unbounded linear operators: domain, range, graph, kernel, resolvent
Characterization of positive operators by means of the resolvent operators
Symmetric and self adjoint operators
Characterization of positive self adjoint operators
Maximality of self adjoint operators with respect to graph inclusion
Positive self adjoint operators on closed subspaces
Their characterization in terms of properties of the resolvent operators
Positive symmetric operator associated with a quadratic form
Semicontinuous quadratic forms and self adjoint operators
Scalar product and norm associated with a quadratic form
Completeness of the norm associated with a semicontinuous quadratic form
Density of the domain of the operator associated with a quadratic form
Construction of a quadratic form from the corresponding operator
Every positive self adjoint operator is associated with a quadratic form
Yosida approximation of operators and Moreau Yosida approximation
13. Convergence of resolvents and G convergence 148
G convergence and convergence in the resolvent sense
F convergence of quadratic forms and G convergence of operators
F convergence of quadratic forms and convergence in the resolvent sense
The case of compact imbedding
G convergence of elliptic operators
x An Introduction to F convergence
14. Increasing set functions 165
Inner and outer regular increasing set functions
Borel and Radon measeres
Inner and outer regular envelopes
Dense and rich sets
Relationships between an increasing set function and its regular envelopes
Subadditivity and superadditivity
A characterization of measures
15. Lower semicontinuous increasing functionals 174
Punctionals depending also on sets
Inner and outer regular envelopes
Lower semicontinuous envelopes
Lower semicontinuous increasing functionals and their envelopes
Moreau Yosida approximation of lower semicontinuous increasing functionals
Local functionals
16. F convergence of increasing set functionals 181
F convergence and connection with F convergence
Urysohn property of F convergence
Compactness properties of F convergence
F convergence and convergence of Moreau Yosida approximations
F limit of a sequence of superadditive increasing functionals
F limit of a sequence of increasing local functionals
17. The topology of F convergence 189
Topologies on the space of lower semicontinuous increasing functionals
Relationships with the topologies introduced in Chapter 10
Connection of these topologies with the problem of convergence of minima
Compactness properties of these topologies
Relationships between these topologies and F convergence
Topologizability of F convergence in the case of k spaces
Lack of separation properties of the topology of F convergence
Metrizability of F convergence under some equi coerciveness conditions
18. The fundamental estimate 202
The fundamental estimate and subadditivity of F limits
F convergence, F convergence, and the fundamental estimate
19. Local functionals and the fundamental estimate 208
Integral functionals satisfying the fundamental estimate
Local functionals satisfying the fundamental estimate
A F compact class of measures
A F compact class of measures
Lower order terms
20. Integral representation of r limits 215
Integral representation theorem for local functionals
A F compact class of integral functionals
Contents xi
21. Boundary conditions 223
F convergence of functional including Dirichlet boundary conditions
Convergence of minima and of minimizers of boundary value problems
22. G convergence of elliptic operators 229
F limits of quadratic integral functional
Compactness of the G convergence of elliptic operators
G convergence of elliptic operators and F convergence
Local character of G convergence
G convergence and non homogeneous Dirichlet problems
G convergence and Neumann problems
23. Translation invariant functionals 239
Jensen Inequality
Approximation by convolutions
Integral representation theorem for translation invariant local functionals
Translation invariant local functionals and affine functions
24. Homogenization 247
Homogenization theorem for integral functionals
The non coercive case
The coercive case
25. Some examples in homogenization 256
Euler conditions for the auxiliary problem
An example in dimension one
Homogenization of elliptic operators
A case of separation of variables
Homogenization of layered materials
Guide to the literature 265
Bibliography 289
Index 339 |
| any_adam_object | 1 |
| author | Dal Maso, Gianni 1954- |
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| author_facet | Dal Maso, Gianni 1954- |
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| author_sort | Dal Maso, Gianni 1954- |
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| dewey-ones | 515 - Analysis |
| dewey-raw | 515/.64 |
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| dewey-tens | 510 - Mathematics |
| discipline | Mathematik |
| format | Book |
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| indexdate | 2024-09-17T14:07:25Z |
| institution | BVB |
| isbn | 081763679X 376433679X 9780817636791 |
| language | English |
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| physical | xiv, 340 Seiten |
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| series2 | Progress in nonlinear differential equations and their applications |
| spelling | Dal Maso, Gianni 1954- (DE-588)11262894X aut An introduction to gamma-convergence Gianni Dal Maso Boston [u.a.] Birkhäuser [1993] © 1993 xiv, 340 Seiten txt rdacontent n rdamedia nc rdacarrier Progress in nonlinear differential equations and their applications Volume 8 Literaturverz. Seite [289] - 337 Homogenisierungsmethode (DE-588)4257770-6 gnd rswk-swf Gamma-Konvergenz (DE-588)4311219-5 gnd rswk-swf Grenzwert (DE-588)4129627-8 gnd rswk-swf Konvergenz (DE-588)4032326-2 gnd rswk-swf Variationsrechnung (DE-588)4062355-5 gnd rswk-swf Gamma-Konvergenz (DE-588)4311219-5 s Konvergenz (DE-588)4032326-2 s DE-604 Homogenisierungsmethode (DE-588)4257770-6 s Variationsrechnung (DE-588)4062355-5 s Grenzwert (DE-588)4129627-8 s Erscheint auch als Online-Ausgabe 978-1-4612-6709-6 Progress in nonlinear differential equations and their applications Volume 8 (DE-604)BV007934389 8 HBZ Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=005092628&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Dal Maso, Gianni 1954- An introduction to gamma-convergence Progress in nonlinear differential equations and their applications Homogenisierungsmethode (DE-588)4257770-6 gnd Gamma-Konvergenz (DE-588)4311219-5 gnd Grenzwert (DE-588)4129627-8 gnd Konvergenz (DE-588)4032326-2 gnd Variationsrechnung (DE-588)4062355-5 gnd |
| subject_GND | (DE-588)4257770-6 (DE-588)4311219-5 (DE-588)4129627-8 (DE-588)4032326-2 (DE-588)4062355-5 |
| title | An introduction to gamma-convergence |
| title_auth | An introduction to gamma-convergence |
| title_exact_search | An introduction to gamma-convergence |
| title_full | An introduction to gamma-convergence Gianni Dal Maso |
| title_fullStr | An introduction to gamma-convergence Gianni Dal Maso |
| title_full_unstemmed | An introduction to gamma-convergence Gianni Dal Maso |
| title_short | An introduction to gamma-convergence |
| title_sort | an introduction to gamma convergence |
| topic | Homogenisierungsmethode (DE-588)4257770-6 gnd Gamma-Konvergenz (DE-588)4311219-5 gnd Grenzwert (DE-588)4129627-8 gnd Konvergenz (DE-588)4032326-2 gnd Variationsrechnung (DE-588)4062355-5 gnd |
| topic_facet | Homogenisierungsmethode Gamma-Konvergenz Grenzwert Konvergenz Variationsrechnung |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=005092628&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
| volume_link | (DE-604)BV007934389 |
| work_keys_str_mv | AT dalmasogianni anintroductiontogammaconvergence |