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Analysis and Numerics of Partial Differential Equations pp 63–115Cite as

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

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Part of the book series: Springer INdAM Series ((SINDAMS,volume 4))

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.

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Acknowledgements

The authors acknowledge the support of the ERC ADG GeMeThNES and the PRIN08-grant from MIUR for the project Optimal transport theory, geometric and functional inequalities, and applications.

The authors also thank A. Mondino for his careful reading of a preliminary version of this manuscript.

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Authors and Affiliations

  1. Scuola Normale Superiore, Piazza dei Cavalieri 7, 56126, Pisa, Italy

    Luigi Ambrosio

  2. Université de Nice, Mathématiques, Parc Valrose, 06108, Nice, France

    Nicola Gigli

  3. Dipartimento di Matematica “F. Casorati”, Università di Pavia, via Ferrata 1, 27100, Pavia, Italy

    Giuseppe Savaré

Authors
  1. Luigi Ambrosio
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  2. Nicola Gigli
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  3. Giuseppe Savaré
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Corresponding author

Correspondence to Giuseppe Savaré .

Editor information

Editors and Affiliations

  1. Ist. Matematica Applicata e, Tecnologie Informatiche (IMATI), CNR Pavia, Via Ferrata 1, Pavia, 27100, Italy

    Franco Brezzi

  2. Dipartimento di Matematica "F. Casorati", Università degli Studi di Pavia, via Ferrata 1, Pavia, 27100, Pavia, Italy

    Piero Colli Franzone

  3. Dipartimento di Matematica "F. Casorati", Università degli Studi di Pavia, Via Ferrata 1, Pavia, 27100, Italy

    Ugo Gianazza

  4. Dipartimento di Matematica "F. Casorati", Università degli Studi di Pavia, via Ferrata 1, Pavia, 27100, Pavia, Italy

    Gianni Gilardi

Additional information

To the memory of Enrico Magenes, whose exemplar life, research and teaching shaped generations of mathematicians.

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Ambrosio, L., Gigli, N., Savaré, G. (2013). Heat Flow and Calculus on Metric Measure Spaces with Ricci Curvature Bounded Below—The Compact Case. In: Brezzi, F., Colli Franzone, P., Gianazza, U., Gilardi, G. (eds) Analysis and Numerics of Partial Differential Equations. Springer INdAM Series, vol 4. Springer, Milano. https://doi.org/10.1007/978-88-470-2592-9_8

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