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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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