Gamma-Convergence for Beginners

by Braides, Andrea

• ISBN13: 9780198507840
• ISBN10: 0198507844
• Format: Hardcover
• Copyright: 2002-09-26
• Publisher: Clarendon Press

Summary

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

Preface	p. vii
Introduction	p. 1
Why a variational convergence?	p. 1
Parade of examples	p. 4
A maieutic approach to [Gamma]-convergence. Direct methods	p. 15
[Gamma]-convergence by numbers	p. 19
Some preliminaries	p. 19
Lower and upper limits	p. 19
Lower semicontinuity	p. 21
[Gamma]-convergence	p. 22
Some examples on the real line	p. 25
The many definitions of [Gamma]-convergence	p. 26
Convergence of minima	p. 28
Upper and lower [Gamma]-limits	p. 30
The importance of being lower semicontinuous	p. 32
Lower semicontinuity of [Gamma]-limits	p. 32
The lower-semicontinuous envelope. Relaxation	p. 32
Approximation of lower-semicontinuous functions	p. 33
The direct method	p. 34
More properties of [Gamma]-limits	p. 34
[Gamma]-limits of monotone sequences	p. 35
Compactness of [Gamma]-convergence	p. 35
[Gamma]-convergence by subsequences	p. 36
[Gamma]-limits indexed by a continuous parameter	p. 37
Development by [Gamma]-convergence	p. 37
Exercises	p. 38
Comments on Chapter 1	p. 39
Integral problems	p. 40
Problems on Lebesgue spaces	p. 40
Weak convergences	p. 41
Weak-coerciveness conditions	p. 43
Weak lower semicontinuity conditions: convexity	p. 44
Relaxation and [Gamma]-convergence in L[superscript p] spaces	p. 47
Problems on Sobolev spaces	p. 50
Weak convergence in Sobolev spaces	p. 50
Integral functionals on Sobolev spaces. Coerciveness conditions	p. 51
Weak lower semicontinuity conditions	p. 52
[Gamma]-convergence and convex analysis	p. 54
Addition of boundary data	p. 57
Some examples with degenerate growth conditions	p. 58
Degeneracy of lower bounds: discontinuities	p. 58
Degeneracy of upper bounds: functionals of the sup norm	p. 59
Exercises	p. 61
Comments on Chapter 2	p. 62
Some homogenization problems	p. 63
A direct approach	p. 63
Different homogenization formulas	p. 66
Limits of oscillating Riemannian metrics	p. 68
Homogenization of Hamilton Jacobi equations	p. 71
Exercises	p. 74
Comments on Chapter 3	p. 75
From discrete systems to integral functionals	p. 76
Discrete functionals	p. 77
Continuous limits	p. 78
Nearest-neighbour interactions: a convexification principle	p. 78
Next-to-nearest neighbour interactions: non-convex relaxation	p. 80
Long-range interactions: homogenization	p. 82
Convergence of minimum problems	p. 84
Exercises	p. 84
Comments on Chapter 4	p. 84
Segmentation problems	p. 85
Model problems	p. 86
The space of piecewise-constant functions	p. 87
Coerciveness conditions	p. 87
Functionals on piecewise-constant functions	p. 88
Lower semicontinuity conditions: subadditivity	p. 88
Relaxation and [Gamma]-convergence	p. 91
Translation-invariant functionals	p. 91
Properties of subadditive functions on R	p. 92
Relaxation: subadditive envelopes	p. 93
[Gamma]-convergence	p. 97
Boundary values	p. 98
Exercises	p. 99
Comments on Chapter 5	p. 100
Caccioppoli partitions	p. 100
Phase-transition problems	p. 102
Phase transitions as segmentation problems	p. 102
Gradient theory for phase-transition problems	p. 103
Gradient theory as a development by [Gamma]-convergence	p. 109
Comments on Chapter 6	p. 112
Free-discontinuity problems	p. 114
Piecewise-Sobolev functions	p. 114
Some model problems	p. 114
Signal reconstruction: the Mumford-Shah functional	p. 115
Fracture mechanics: the Griffith functional	p. 115
Functionals on piecewise-Sobolev functions	p. 116
Examples of existence results	p. 117
Comments on Chapter 7	p. 119
Special functions of bounded variation	p. 120
Approximation of free-discontinuity problems	p. 121
The Ambrosio Tortorelli approximation	p. 121
Approximation by convolution problems	p. 124
Convolution integral functionals	p. 125
Limits of convolution functionals	p. 126
Finite-difference approximation	p. 130
Comments on Chapter 8	p. 131
More homogenization problems	p. 132
Oscillations and phase transitions	p. 132
Phase accumulation	p. 135
Homogenization of free-discontinuity problems	p. 137
Comments on Chapter 9	p. 138
Interaction between elliptic problems and partition problems	p. 139
Quantitative conditions for lower semicontinuity	p. 139
Existence without lower semicontinuity	p. 142
Relaxation by interaction	p. 143
Exercises	p. 148
Comments on Chapter 10	p. 148
Structured deformations	p. 149
Discrete systems and free-discontinuity problems	p. 150
Interpolation with piecewise-Sobolev functions	p. 151
Equivalent energies on piecewise-Sobolev functions	p. 153
Softening and fracture problems as limits of discrete models	p. 154
Fracture as a phase transition	p. 156
Malik Perona approximation of free-discontinuity problems	p. 159
Exercises	p. 159
Comments on Chapter 11	p. 160
Some comments on vectorial problems	p. 161
Lower semicontinuity conditions	p. 162
Quasiconvexity	p. 163
Convexity and polyconvexity	p. 164
Homogenization and convexity conditions	p. 165
Instability of polyconvexity	p. 166
Density of isotropic quadratic forms	p. 168
Comments on Chapter 12	p. 169
Dirichlet problems in perforated domains	p. 171
Statement of the [Gamma]-convergence result	p. 172
A joining lemma on varying domains	p. 174
Proof of the lim inf inequality	p. 177
Proof of the lim sup inequality	p. 178
Comments on Chapter 13	p. 181
Dimension-reduction problems	p. 182
Convex energies	p. 182
Non-convex vector-valued problems	p. 185
Comments on Chapter 14	p. 186
The 'slicing' method	p. 187
A lower inequality by the slicing method	p. 188
An upper inequality by density	p. 191
Comments on Chapter 15	p. 193
An introduction to the localization method of [Gamma]-convergence	p. 194
Appendices	p. 197
Some quick recalls	p. 197
Convexity	p. 197
Sobolev spaces	p. 198
Sets of finite perimeter	p. 200
Characterization of [Gamma]-convergence for 1D integral problems	p. 203
List of symbols	p. 207
References	p. 209
Index	p. 217
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