Gradient Flows

, by Ambrosio, Luigi; Gigli, Nicola; Savare, Giuseppe

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ISBN: 9783764387211 | 3764387211
Cover: Paperback
Copyright: 6/3/2008

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This book is devoted to a theory of gradient flows in spaces which are not necessarily endowed with a natural linear or differentiable structure. It consists of two parts, the first one concerning gradient flows in metric spaces and the second one devoted to gradient flows in the space of probability measures on a separable Hilbert space, endowed with the Kantorovich-Rubinstein-Wasserstein distance.The two parts have some connections, due to the fact that the space of probability measures provides an important model to which the "metric" theory applies, but the book is conceived in such a way that the two parts can be read independently, the first one by the reader more interested in non-smooth analysis and analysis in metric spaces, and the second one by the reader more orientated towards the applications in partial differential equations, measure theory and probability.

Preface to the Second Edition	p. ix
Introduction	p. 1
Notation	p. 18
Gradient Flow in Metric Spaces	p. 21
Curves and Gradients in Metric Spaces	p. 23
Absolutely continuous curves and metric derivative	p. 23
Upper gradients	p. 26
Curves of maximal slope	p. 30
Curves of maximal slope in Hilbert and Banach spaces	p. 32
Existence of Curves of Maximal Slope	p. 39
Main topological assumptions	p. 42
Solvability of the discrete problem and compactness of discrete trajectories	p. 44
Generalized minimizing movements and curves of maximal slope	p. 45
The (geodesically) convex case	p. 49
Proofs of the Convergence Theorems	p. 59
Moreau-Yosida approximation	p. 59
A priori estimates for the discrete Solutions	p. 66
A compactness argument	p. 69
Conclusion of the proofs of the convergence theorems	p. 71
Generation of Contraction Semigroups	p. 75
Cauchy-type estimates for discrete Solutions	p. 82
Discrete variational inequalities	p. 82
Piecewise affine Interpolation and comparison results	p. 84
Convergence of discrete Solutions	p. 89
Convergence when the initial datum u₀ ∈ <$>D(\phi)<$>	p. 89
Convergence when the initial datum u₀ ∈ <$>\overline{D(\phi)}<$>	p. 92
Regularizing effect, uniqueness and the semigroup property	p. 93
Optimal error estimates	p. 97
The case ¿ = 0	p. 97
The case ¿ &neq; 0	p. 99
Gradient Flow in the Space of Probability Measures	p. 103
Preliminary Results on Measure Theory	p. 105
Narrow convergence, tightness, and uniform integrability	p. 106
Unbounded and l.s.c. integrands	p. 109
Hilbert spaces and weak topologies	p. 113
Transport of measures	p. 118
Measure-valued maps and disintegration theorem	p. 121
Convergence of plans and convergence of maps	p. 124
Approximate differentiability and area formula in Euclidean spaces	p. 128
The Optimal Transportation Problem	p. 133
Optimality conditions	p. 135
Optimal transport maps and their regularity	p. 139
Approximate differentiability of the optimal transport map	p. 142
The infinite dimensional case	p. 147
The quadratic case p = 2	p. 149
The Wasserstein Distance and its Behaviour along Geodesics	p. 151
The Wasserstein distance	p. 151
Interpolation and geodesics	p. 158
The curvature properties of <$>{\cal P}_2(X)<$>	p. 160
A.C. Curves in <$>{\cal P}_p(X)<$> and the Continuity Equation	p. 167
The continuity equation in <$>{\op R}^d<$>	p. 169
A probabilistic representation of Solutions of the continuity equation	p. 178
Absolutely continuous curves in <$>{\cal P}_p(X)<$>	p. 182
The tangent bundle to <$>{\cal P}_p(X)<$>	p. 189
Tangent space and optimal maps	p. 194
Convex Functionals in <$>{\cal P}_p(X)<$>	p. 201
¿-geodesically convex functionals in <$>{\cal P}_p(X)<$>	p. 202
Convexity along generalized geodesics	p. 205
Examples of convex functionals in <$>{\cal P}_p(X)<$>	p. 209
Relative entropy and convex functionals of measures	p. 215
Log-concavity and displacement convexity	p. 220
Metric Slope and Subdifferential Calculus in <$>{\cal P}_p(X)<$>	p. 227
Subdifferential calculus in <$>{\cal P}_2^r(X)<$>: the regular case	p. 229
The case of ¿-convex functionals along geodesics	p. 231
Regular functionals	p. 232
Differentiability properties of the p-Wasserstein distance	p. 234
Subdifferential calculus in <$>{\cal P}_p(X)<$>: the general case	p. 240
The case of ¿-convex functionals along geodesics	p. 244
Regular functionals	p. 246
Example of subdifferentials	p. 254
Variational integrals: the smooth case	p. 254
The potential energy	p. 255
The internal energy	p. 257
The relative internal energy	p. 265
The interaction energy	p. 267
The opposite Wasserstein distance	p. 269
The sum of internal, potential and interaction energy	p. 272
Relative entropy and Fisher Information in infinite dimensions	p. 276
Gradient Flows and Curves of Maximal Slope in <$>{\cal P}_p(X)<$>	p. 279
The gradient flow equation and its metric formulations	p. 280
Gradient flows and curves of maximal slope	p. 283
Gradient flows for ¿-convex functionals	p. 284
The convergence of the "Minimizing Movement" scheme	p. 286
Gradient flows for ¿-convex functionals along generalized geodesics	p. 295
Applications to Evolution PDE's	p. 298
Gradient flows in <$>{\cal P}_p(X)<$> for regular functionals	p. 304
Appendix	p. 307
Carathéodory and normal integrands	p. 307
Weak convergence of plans and disintegrations	p. 308
PC metric spaces and their geometric tangent cone	p. 310
The geometric tangent spaces in <$>{\cal P}_2(X)<$>	p. 314
Bibliography	p. 331
Index	p. 333
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Gradient Flows

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