Heat Flow and Calculus on Metric Measure Spaces with Ricci Curvature Bounded Below—The Compact Case View Full Text


Ontology type: schema:Chapter     


Chapter Info

DATE

2013

AUTHORS

Luigi Ambrosio , Nicola Gigli , Giuseppe Savaré

ABSTRACT

We provide a quick overview of various calculus tools and of the main results concerning the heat flow on compact metric measure spaces, with applications to spaces with lower Ricci curvature bounds. Topics include the Hopf-Lax semigroup and the Hamilton-Jacobi equation in metric spaces, a new approach to differentiation and to the theory of Sobolev spaces over metric measure spaces, the equivalence of the L 2-gradient flow of a suitably defined “Dirichlet energy” and the Wasserstein gradient flow of the relative entropy functional, a metric version of Brenier’s Theorem, and a new (stronger) definition of Ricci curvature bound from below for metric measure spaces. This new notion is stable w.r.t. measured Gromov-Hausdorff convergence and it is strictly connected with the linearity of the heat flow. More... »

PAGES

63-115

Book

TITLE

Analysis and Numerics of Partial Differential Equations

ISBN

978-88-470-2591-2
978-88-470-2592-9

Identifiers

URI

http://scigraph.springernature.com/pub.10.1007/978-88-470-2592-9_8

DOI

http://dx.doi.org/10.1007/978-88-470-2592-9_8

DIMENSIONS

https://app.dimensions.ai/details/publication/pub.1051961221


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