𝔖 Bobbio Scriptorium
✦   LIBER   ✦

Gradient flows: In metric spaces and in the space of probability measures

✍ Scribed by Luigi Ambrosio, Nicola Gigli, Giuseppe Savaré

Book ID
127456248
Publisher
Birkhäuser Basel
Year
2008
Tongue
English
Weight
2 MB
Series
Lectures in Mathematics. ETH Zürich
Edition
2nd
Category
Library
ISBN
3764387211

No coin nor oath required. For personal study only.

✦ Synopsis

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.

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We develop the long-time analysis for gradient flow equations in metric spaces. In particular, we consider two notions of solutions for metric gradient flows, namely energy and generalized solutions. While the former concept coincides with the notion of curves of maximal slope of Ambrosio et al. (20