Unbounded functionals in the calculus of variations : representation,relaxation, and homogenization

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Title:

Personal Author:

Carbone, Luciano

Publication Information:

Boca Raton : Chapman & Hall, 2002

ISBN:

9781584882350

Subject Term:

Calculus of variations

Functionals

Added Author:

Arcangelis, Riccardo de

Available:*

Library	Item Barcode	Call Number	Material Type	Item Category 1	Status
Searching... PSZ JB	30000004821520	QA315 C27 2002	Open Access Book	Book	Searching... Unknown

Over the last few decades, research in elastic-plastic torsion theory, electrostatic screening, and rubber-like nonlinear elastomers has pointed the way to some interesting new classes of minimum problems for energy functionals of the calculus of variations. This advanced-level monograph addresses these issues by developing the framework of a general theory of integral representation, relaxation, and homogenization for unbounded functionals.

The first part of the book builds the foundation for the general theory with concepts and tools from convex analysis, measure theory, and the theory of variational convergences. The authors then introduce some function spaces and explore some lower semicontinuity and minimization problems for energy functionals. Next, they survey some specific aspects the theory of standard functionals.

The second half of the book carefully develops a theory of unbounded, translation invariant functionals that leads to results deeper than those already known, including unique extension properties, representation as integrals of the calculus of variations, relaxation theory, and homogenization processes. In this study, some new phenomena are pointed out. The authors' approach is unified and elegant, the text well written, and the results intriguing and useful, not just in various fields of mathematics, but also in a range of applied mathematics, physics, and material science disciplines.

Author Notes

Luciano Carbone and Riccardo De Arcangelis are Full Professors of Mathematical Analysis at the University of Naples "Federico II," Italy.

Preface	p. xi
Basic Notations and Recalls	p. 1
1 Elements of Convex Analysis	p. 9
1.1 Convex Sets and Functions	p. 9
1.2 Convex and Lower Semicontinuous Envelopes in R[superscript n]	p. 23
1.3 Lower Semicontinuous Envelopes of Convex Envelopes	p. 29
1.4 Convex Envelopes of Lower Semicontinuous Envelopes	p. 35
2 Elements of Measure and Increasing Set Functions Theories	p. 45
2.1 Measures and Integrals	p. 45
2.2 Basics on L[superscript p] Spaces	p. 55
2.3 Derivation of Measures	p. 60
2.4 Abstract Measure Theory in Topological Settings	p. 63
2.5 Local Properties of Boundaries of Open Subsets of R[superscript n]	p. 67
2.6 Increasing Set Functions	p. 70
2.7 Increasing Set Functionals	p. 79
3 Minimization Methods and Variational Convergences	p. 83
3.1 The Direct Methods in the Calculus of Variations	p. 83
3.2 [Gamma]-Convergence	p. 87
3.3 Applications to the Calculus of Variations	p. 94
3.4 [Gamma]-Convergence in Topological Vector Spaces and of Increasing Set Functionals	p. 99
3.5 Relaxation	p. 102
4 BV and Sobolev Spaces	p. 107
4.1 Regularization of Measures and of Summable Functions	p. 107
4.2 BV Spaces	p. 113
4.3 Sobolev Spaces	p. 121
4.4 Some Compactness Criteria	p. 130
4.5 Periodic Sobolev Functions	p. 134
5 Lower Semicontinuity and Minimization of Integral Functionals	p. 137
5.1 Functionals on BV Spaces	p. 137
5.2 Functionals on Sobolev Spaces	p. 142
5.3 Minimization of Integral Functionals	p. 144
6 Classical Results and Mathematical Models Originating Unbounded Functionals	p. 149
6.1 Classical Unique Extension Results	p. 149
6.2 Classical Integral Representation Results	p. 150
6.3 Classical Relaxation Results	p. 153
6.4 Classical Homogenization Results	p. 155
6.5 Mathematical Aspects of Some Physical Models Originating Unbounded Functionals	p. 157
7 Abstract Regularization and Jensen's Inequality	p. 159
7.1 Integral of Functions with Values in Locally Convex Topological Vector Spaces	p. 159
7.2 On the Definition of a Functional on Functions and on Their Equivalence Classes	p. 163
7.3 Regularization of Functions in Locally Convex Topological Vector Subspaces of L[superscript 1 subscript loc] (R[superscript n])	p. 165
7.4 Applications to Convex Functionals on BV Spaces	p. 169
8 Unique Extension Results	p. 177
8.1 Unique Extension Results for Inner Regular Functionals	p. 178
8.2 Existence and Uniqueness Results	p. 180
8.3 Unique Extension Results for Measure Like Functionals	p. 182
8.4 Some Applications	p. 186
8.5 A Note on Lavrentiev Phenomenon	p. 189
9 Integral Representation for Unbounded Functionals	p. 191
9.1 Representation on Linear Functions	p. 191
9.2 Representation on Continuously Differentiable Functions	p. 192
9.3 Representation on Sobolev Spaces	p. 199
9.4 Representation on BV Spaces	p. 207
10 Relaxation of Unbounded Functionals	p. 211
10.1 Notations and Elementary Properties of Relaxed Functionals in the Neumann Case	p. 211
10.2 Relaxation of Neumann Problems: the Case of Bounded Effective Domain with Nonempty Interior	p. 214
10.3 Relaxation of Neumann Problems: the Case of Bounded Effective Domain with Empty Interior	p. 219
10.4 Relaxation of Neumann Problems: a First Result without Boundedness Assumptions on the Effective Domain	p. 226
10.5 Relaxation of Neumann Problems: Relaxation in BV Spaces	p. 229
10.6 Notations and Elementary Properties of Relaxed Functionals in the Dirichlet Case	p. 232
10.7 Relaxation of Dirichlet Problems	p. 234
10.8 Applications to Minimum Problems	p. 246
10.9 Additional Remarks on Integral Representation on the Whole Space of Lipschitz Functions	p. 253
11 Cut-off Functions and Partitions of Unity	p. 261
11.1 Cut-off Functions	p. 261
11.2 Partitions of Unity	p. 269
12 Homogenization of Unbounded Functionals	p. 273
12.1 Notations and Basic Results	p. 274
12.2 Some Properties of [Gamma]-Limits	p. 280
12.3 Finiteness Conditions	p. 287
12.4 Representation on Linear Functions	p. 293
12.5 A Blow-up Condition	p. 306
12.6 Representation Results	p. 307
12.7 Applications to the Convergence of Minima and of Minimizers	p. 311
13 Homogenization of Unbounded Functionals with Special Constraints	p. 319
13.1 Homogenization with Fixed Constraints: the Case of Neumann Boundary Conditions	p. 319
13.2 Homogenization with Fixed Constraints: the Case of Dirichlet Boundary Conditions	p. 328
13.3 Homogenization with Fixed Constraints: the Case of Mixed Boundary Conditions	p. 331
13.4 Homogenization with Fixed Constraints: Applications to the Convergence of Minima and of Minimizers	p. 335
13.5 Homogenization with Oscillating Special Constraints	p. 345
13.6 Final Remarks	p. 351
14 Some Explicit Computations of Homogenized Energies in Mathematical Models Originating Unbounded Functionals	p. 353
14.1 Homogenization in Elastic-Plastic Torsion	p. 353
14.2 Homogenization in the Modelling of Nonlinear Elastomers	p. 359
14.3 Homogenization in Electrostatic Screening	p. 364
Bibliography	p. 375
List of Symbols	p. 389
Index	p. 391

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