Gradient Flows: In Metric Spaces and in the Space of Probability MeasuresSpringer Science & Business Media, 29.10.2008 - 334 Seiten The book is devoted to the theory of gradient flows in the general framework of metric spaces, and in the more specific setting of the space of probability measures, which provide a surprising link between optimal transportation theory and many evolutionary PDE's related to (non)linear diffusion. Particular emphasis is given to the convergence of the implicit time discretization method and to the error estimates for this discretization, extending the well established theory in Hilbert spaces. The book is split in two main parts that can be read independently of each other. |
Inhalt
Proofs of the Convergence Theorems | 58 |
Generation of Contraction Semigroups | 75 |
18 | 100 |
Preliminary Results on Measure Theory | 105 |
The Optimal Transportation Problem 133 | 132 |
The Wasserstein Distance and its Behaviour along Geodesics | 151 |
A C Curves in PpX and the Continuity Equation | 167 |
30 | 182 |
32 | 220 |
Metric Slope and Subdifferential Calculus in PpX 227 | 229 |
39 | 251 |
45 | 267 |
59 | 273 |
Gradient Flows and Curves of Maximal Slope in PpX | 279 |
Appendix | 307 |
| 331 | |
Andere Ausgaben - Alle anzeigen
Gradient Flows: In Metric Spaces and in the Space of Probability Measures Luigi Ambrosio,Nicola Gigli,Giuseppe Savare Keine Leseprobe verfügbar - 2009 |
Gradient Flows: In Metric Spaces and in the Space of Probability Measures Luigi Ambrosio,Nicola Gigli,Giuseppe Savare Keine Leseprobe verfügbar - 2009 |
Häufige Begriffe und Wortgruppen
absolutely continuous apply approximation argument assume assumptions Banach space belongs Borel bounded Chapter characterization choose compact condition consider constant convex curve defined Definition denote derivative differentiable distance du(a equation equivalent estimate Example exists fact finite follows formula function geodesics given gives gradient flow H-co Hilbert space holds induced inequality integrable introduce Lemma limit lower semicontinuous maximal measure minimal Moreover narrow narrowly converging Notice Observe obtain optimal transport map particular Proof Proposition prove provides recall regular relative Remark respect result satisfies separable sequence simply ſlº slope ſº solution strong subdifferential subset ſun suppose taking tangent Theorem tight To(u uniformly unique variational vector field Wasserstein yields