| Introduction |
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1 | (6) |
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Functions of bounded variation |
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7 | (20) |
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Measure theory. Basic notation |
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7 | (3) |
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Supremum of a family of measures |
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9 | (1) |
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Construction of measures. Hausdorff measures |
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10 | (2) |
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Caratheodory's construction |
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10 | (1) |
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11 | (1) |
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The De Giorgi and Letta measure criterion |
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11 | (1) |
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Weak convergence of measures |
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12 | (2) |
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Weak convergence of measures as set functions |
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13 | (1) |
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14 | (1) |
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14 | (2) |
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BV functions of one variable |
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16 | (1) |
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16 | (2) |
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Structure of the sets of finite perimeter |
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18 | (1) |
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19 | (1) |
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Structure of BV functions |
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20 | (3) |
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1-dimensional sections of BV functions |
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21 | (1) |
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22 | (1) |
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23 | (4) |
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Special functions of bounded variation |
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27 | (12) |
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SBV functions. A compactness theorem |
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27 | (2) |
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General lower semicontinuity conditions in one dimension |
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29 | (6) |
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34 | (1) |
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A lower semicontinuity theorem in higher dimensions |
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35 | (1) |
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The Mumford-Shah functional |
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36 | (3) |
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36 | (2) |
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The Mumford-Shah functional |
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38 | (1) |
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Examples of approximation |
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39 | (48) |
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Γ-convergence: an overview |
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39 | (3) |
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42 | (14) |
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Approximation of the perimeter by elliptic functionals |
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42 | (4) |
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46 | (1) |
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Approximation of the Mumford-Shah functional by elliptic functionals |
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47 | (4) |
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Approximation of free-discontinuity problems by elliptic functionals |
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51 | (5) |
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Approximations by high-order perturbations |
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56 | (11) |
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Surface energies generated by high-order singular perturbation |
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56 | (7) |
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63 | (1) |
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Approximation of the Mumford-Shah functional by high-order perturbations |
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64 | (3) |
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67 | (1) |
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67 | (11) |
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Non-local approximation of the Mumford-Shah functional |
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67 | (4) |
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71 | (1) |
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Non-local approximation of free-discontinuity problems |
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72 | (5) |
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77 | (1) |
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Finite-difference approximation of free-discontinuity problems |
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78 | (9) |
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85 | (2) |
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A general approach to approximation |
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87 | (16) |
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A lower inequality by slicing |
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87 | (9) |
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88 | (1) |
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A lower estimate for the perimeter approximation |
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89 | (3) |
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A lower estimate for the elliptic approximation |
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92 | (2) |
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A lower estimate for the approximation by high-order perturbations |
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94 | (2) |
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An upper inequality by density |
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96 | (5) |
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An upper estimate for the perimeter approximation |
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97 | (2) |
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99 | (1) |
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An upper estimate for the elliptic approximation |
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100 | (1) |
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101 | (2) |
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103 | (28) |
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Non-local approximation of the Mumford-Shah functional |
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103 | (21) |
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Estimate from below of the volume term |
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103 | (5) |
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Estimate from below of the surface term |
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108 | (6) |
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Estimate from below of the Γ-limit |
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114 | (4) |
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Estimate from above of the Γ-limit |
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118 | (1) |
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119 | (5) |
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Finite-difference approximation of the Mumford-Shah functional |
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124 | (7) |
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127 | (2) |
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129 | (1) |
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130 | (1) |
| Appendix |
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131 | (12) |
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131 | (3) |
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B Approximation of polyhedral energies |
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134 | (5) |
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C An integral representation result |
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139 | (2) |
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141 | (2) |
| Notation |
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143 | (2) |
| References |
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145 | (4) |
| Index |
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149 | |
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