Details

Gradient Flows

In Metric Spaces and in the Space of Probability Measures
Lectures in Mathematics. ETH Zürich

von: Luigi Ambrosio, Nicola Gigli, Giuseppe Savare
CHF 60.00
Verlag:	Birkhäuser
Format:	PDF
Veröffentl.:	30.03.2006
ISBN/EAN:	9783764373092
Sprache:	englisch
Anzahl Seiten:	340

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Dieses eBook enthält ein Wasserzeichen.

Beschreibungen

Titelbeschreibung

This book is devoted to a theory of gradient ?ows in spaces which are not nec- sarily endowed with a natural linear or di?erentiable structure. It is made of two parts, the ?rst one concerning gradient ?ows in metric spaces and the second one 2 1 devoted to gradient ?ows in the L -Wasserstein space of probability measures on p a separable Hilbert space X (we consider the L -Wasserstein distance, p? (1,?), as well). The two parts have some connections, due to the fact that the Wasserstein 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 ?rst one by the reader more interested to Non-Smooth Analysis and Analysis in Metric Spaces, and the second one by the reader more oriented to theapplications in Partial Di?erential Equations, Measure Theory and Probability.

Inhaltsverzeichnis

Gradient Flow in Metric Spaces.- Curves and Gradients in Metric Spaces.- Existence of Curves of Maximal Slope and their Variational Approximation.- Proofs of the Convergence Theorems.- Uniqueness, Generation of Contraction Semigroups, Error Estimates.- Notation.- Gradient Flow in the Space of Probability Measures.- Preliminary Results on Measure Theory.- The Optimal Transportation Problem.- The Wasserstein Distance and its Behaviour along Geodesics.- Absolutely Continuous Curves in Pp(X) and the Continuity Equation.- Convex Functionals in Pp(X).- Metric Slope and Subdifferential Calculus in Pp(X).- Gradient Flows and Curves of Maximal Slope in Pp(X).

Sonderbeitrag

Originating from lectures by L. Ambrosio at the ETH Zürich in Fall 2001 Substantially extended and revised in cooperation with the co-authors Serves as textbook and reference book on the topic Presented as much as possible in a self-contained way Containing new results that never appeared elsewhere

Verkaufsargumente

<P>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.<BR>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.</P>