Nonlinear Diffusion Equations and Curvature Conditions in Metric Measure Spaces

The aim of this paper is to provide new characterizations of the curvature dimension condition in the context of metric measure spaces (X,\mathsf d,\mathfrak m). On the geometric side, the authors' new approach takes into account suitable weighted action functionals which provide the natural mo...

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Bibliographic Details
Main Authors: Ambrosio, Luigi, Mondino, Andrea, Savaré, Giuseppe
Format: eBook Book
Language:English
Published: Providence, Rhode Island American Mathematical Society 2019
Edition:1
Series:Memoirs of the American Mathematical Society
Subjects:
Metric spaces
Bakry-Émery tensor
Ricci curvature
Nonlinear diffusion
Optimal transport
Metric measure spaces
Nonsmooth Riemannian geometry
Displacement convexity
ISBN:9781470439132, 1470439131
ISSN:0065-9266, 1947-6221
Online Access:Get full text
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Summary:The aim of this paper is to provide new characterizations of the curvature dimension condition in the context of metric measure spaces (X,\mathsf d,\mathfrak m). On the geometric side, the authors' new approach takes into account suitable weighted action functionals which provide the natural modulus of K-convexity when one investigates the convexity properties of N-dimensional entropies. On the side of diffusion semigroups and evolution variational inequalities, the authors' new approach uses the nonlinear diffusion semigroup induced by the N-dimensional entropy, in place of the heat flow. Under suitable assumptions (most notably the quadraticity of Cheeger's energy relative to the metric measure structure) both approaches are shown to be equivalent to the strong \mathrm {CD}^{*}(K,N) condition of Bacher-Sturm.
Bibliography:Includes bibliographical reference (p. 119-121)
November 2019, volume 262, number 1270 (seventh of 7 numbers)
ISBN:9781470439132
1470439131
ISSN:0065-9266
1947-6221
DOI:10.1090/memo/1270