Gamma-Convergence for Beginners

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  • Format: Hardcover
  • Copyright: 2002-09-26
  • Publisher: Clarendon Press

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The theory of Gamma-convergence is commonly recognized as an ideal and flexible tool for the description of the asymptotic behaviour of variational problems. Its applications range from the mathematical analysis of composites to the theory of phase transitions, from Image Processing toFracture Mechanics. This text, written by an expert in the field, provides a brief and simple introduction to this subject, based on the treatment of a series of fundamental problems that illustrate the main features and techniques of Gamma-convergence and at the same time provide a stimulatingstarting point for further studies. The main part is set in a one-dimensional framework that highlights the main issues of Gamma-convergence without the burden of higher-dimensional technicalities. The text deals in sequence with increasingly complex problems, first treating integral functionals,then homogenisation, segmentation problems, phase transitions, free-discontinuity problems and their discrete and continuous approximation, making stimulating connections among those problems and with applications. The final part is devoted to an introduction to higher-dimensional problems, wheremore technical tools are usually needed, but the main techniques of Gamma-convergence illustrated in the previous section may be applied unchanged. The book and its structure originate from the author's experience in teaching courses on this subject to students at PhD level in all fields of Applied Analysis, and from the interaction with many specialists in Mechanics and Computer Vision, which have helped in making the text addressed also to anon-mathematical audience. The material of the book is almost self-contained, requiring only some basic notion of Measure Theory and Functional Analysis.

Author Biography

Andrea Braides is a Professor in the Department of Mathematics, Universita di Roma 'Tor Vergata'.

Table of Contents

Prefacep. vii
Introductionp. 1
Why a variational convergence?p. 1
Parade of examplesp. 4
A maieutic approach to [Gamma]-convergence. Direct methodsp. 15
[Gamma]-convergence by numbersp. 19
Some preliminariesp. 19
Lower and upper limitsp. 19
Lower semicontinuityp. 21
[Gamma]-convergencep. 22
Some examples on the real linep. 25
The many definitions of [Gamma]-convergencep. 26
Convergence of minimap. 28
Upper and lower [Gamma]-limitsp. 30
The importance of being lower semicontinuousp. 32
Lower semicontinuity of [Gamma]-limitsp. 32
The lower-semicontinuous envelope. Relaxationp. 32
Approximation of lower-semicontinuous functionsp. 33
The direct methodp. 34
More properties of [Gamma]-limitsp. 34
[Gamma]-limits of monotone sequencesp. 35
Compactness of [Gamma]-convergencep. 35
[Gamma]-convergence by subsequencesp. 36
[Gamma]-limits indexed by a continuous parameterp. 37
Development by [Gamma]-convergencep. 37
Exercisesp. 38
Comments on Chapter 1p. 39
Integral problemsp. 40
Problems on Lebesgue spacesp. 40
Weak convergencesp. 41
Weak-coerciveness conditionsp. 43
Weak lower semicontinuity conditions: convexityp. 44
Relaxation and [Gamma]-convergence in L[superscript p] spacesp. 47
Problems on Sobolev spacesp. 50
Weak convergence in Sobolev spacesp. 50
Integral functionals on Sobolev spaces. Coerciveness conditionsp. 51
Weak lower semicontinuity conditionsp. 52
[Gamma]-convergence and convex analysisp. 54
Addition of boundary datap. 57
Some examples with degenerate growth conditionsp. 58
Degeneracy of lower bounds: discontinuitiesp. 58
Degeneracy of upper bounds: functionals of the sup normp. 59
Exercisesp. 61
Comments on Chapter 2p. 62
Some homogenization problemsp. 63
A direct approachp. 63
Different homogenization formulasp. 66
Limits of oscillating Riemannian metricsp. 68
Homogenization of Hamilton Jacobi equationsp. 71
Exercisesp. 74
Comments on Chapter 3p. 75
From discrete systems to integral functionalsp. 76
Discrete functionalsp. 77
Continuous limitsp. 78
Nearest-neighbour interactions: a convexification principlep. 78
Next-to-nearest neighbour interactions: non-convex relaxationp. 80
Long-range interactions: homogenizationp. 82
Convergence of minimum problemsp. 84
Exercisesp. 84
Comments on Chapter 4p. 84
Segmentation problemsp. 85
Model problemsp. 86
The space of piecewise-constant functionsp. 87
Coerciveness conditionsp. 87
Functionals on piecewise-constant functionsp. 88
Lower semicontinuity conditions: subadditivityp. 88
Relaxation and [Gamma]-convergencep. 91
Translation-invariant functionalsp. 91
Properties of subadditive functions on Rp. 92
Relaxation: subadditive envelopesp. 93
[Gamma]-convergencep. 97
Boundary valuesp. 98
Exercisesp. 99
Comments on Chapter 5p. 100
Caccioppoli partitionsp. 100
Phase-transition problemsp. 102
Phase transitions as segmentation problemsp. 102
Gradient theory for phase-transition problemsp. 103
Gradient theory as a development by [Gamma]-convergencep. 109
Comments on Chapter 6p. 112
Free-discontinuity problemsp. 114
Piecewise-Sobolev functionsp. 114
Some model problemsp. 114
Signal reconstruction: the Mumford-Shah functionalp. 115
Fracture mechanics: the Griffith functionalp. 115
Functionals on piecewise-Sobolev functionsp. 116
Examples of existence resultsp. 117
Comments on Chapter 7p. 119
Special functions of bounded variationp. 120
Approximation of free-discontinuity problemsp. 121
The Ambrosio Tortorelli approximationp. 121
Approximation by convolution problemsp. 124
Convolution integral functionalsp. 125
Limits of convolution functionalsp. 126
Finite-difference approximationp. 130
Comments on Chapter 8p. 131
More homogenization problemsp. 132
Oscillations and phase transitionsp. 132
Phase accumulationp. 135
Homogenization of free-discontinuity problemsp. 137
Comments on Chapter 9p. 138
Interaction between elliptic problems and partition problemsp. 139
Quantitative conditions for lower semicontinuityp. 139
Existence without lower semicontinuityp. 142
Relaxation by interactionp. 143
Exercisesp. 148
Comments on Chapter 10p. 148
Structured deformationsp. 149
Discrete systems and free-discontinuity problemsp. 150
Interpolation with piecewise-Sobolev functionsp. 151
Equivalent energies on piecewise-Sobolev functionsp. 153
Softening and fracture problems as limits of discrete modelsp. 154
Fracture as a phase transitionp. 156
Malik Perona approximation of free-discontinuity problemsp. 159
Exercisesp. 159
Comments on Chapter 11p. 160
Some comments on vectorial problemsp. 161
Lower semicontinuity conditionsp. 162
Quasiconvexityp. 163
Convexity and polyconvexityp. 164
Homogenization and convexity conditionsp. 165
Instability of polyconvexityp. 166
Density of isotropic quadratic formsp. 168
Comments on Chapter 12p. 169
Dirichlet problems in perforated domainsp. 171
Statement of the [Gamma]-convergence resultp. 172
A joining lemma on varying domainsp. 174
Proof of the lim inf inequalityp. 177
Proof of the lim sup inequalityp. 178
Comments on Chapter 13p. 181
Dimension-reduction problemsp. 182
Convex energiesp. 182
Non-convex vector-valued problemsp. 185
Comments on Chapter 14p. 186
The 'slicing' methodp. 187
A lower inequality by the slicing methodp. 188
An upper inequality by densityp. 191
Comments on Chapter 15p. 193
An introduction to the localization method of [Gamma]-convergencep. 194
Appendicesp. 197
Some quick recallsp. 197
Convexityp. 197
Sobolev spacesp. 198
Sets of finite perimeterp. 200
Characterization of [Gamma]-convergence for 1D integral problemsp. 203
List of symbolsp. 207
Referencesp. 209
Indexp. 217
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