Nonlinear Diffusion Equations and Curvature Conditions in Metric Measure Spaces
Published on 30. March 2020
Book
Paperback/Softback
121 pages
978-1-4704-3913-2 (ISBN)
Description
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.
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.
More details
Series
Language
English
Place of publication
Providence
United States
Target group
Professional and scholarly
Dimensions
Height: 254 mm
Width: 178 mm
Weight
250 gr
ISBN-13
978-1-4704-3913-2 (9781470439132)
Copyright in bibliographic data and cover images is held by Nielsen Book Services Limited or by the publishers or by their respective licensors: all rights reserved.
Schweitzer Classification
Persons
Luigi Ambrosio, Scuola Normale Superiore, Pisa, Italy.
Andrea Mondino, University of Warwick, Coventry, United Kingdom.
Giuseppe Savare, Universita di Pavia, Italy.
Andrea Mondino, University of Warwick, Coventry, United Kingdom.
Giuseppe Savare, Universita di Pavia, Italy.
Content
Introduction
Contraction and Convexity via Hamiltonian Estimates: An Heuristic Argument
Part I. Nonlinear Diffusion Equations and Their Linearization in Dirichlet Spaces: Dirichlet Forms, Homogeneous Spaces and Nonlinear Diffusion
Backward and Forward Linearizations of Nonlinear Diffusion
Part II. Continuity Equation and Curvature Conditions in Metric Measure Spaces: Preliminaries
Absolutely Continuous Curves in Wasserstein Spaces and Continuity Inequalities in a Metric Setting
Weighted Energy Functionals along Absolutely Continuous Curves
Dynamic Kantorovich Potentials, Continuity Equation and Dual Weighted Cheeger Energies
The $\mathrm{RCDS}^{*}(K, N)$ Condition and Its Characterizations through Weighted Convexity and Evolution Variational Inequalities
Part III. Bakry-Emery Condition and Nonlinear Diffusion: The Bakry-Emery Condition
Nonlinear Diffusion Equations and Action Estimates
The Equivalence Between $\mathrm{BE}(K, N)$ and $\mathrm{RCDS}^{*}(K, N)$
Bibliography.
Contraction and Convexity via Hamiltonian Estimates: An Heuristic Argument
Part I. Nonlinear Diffusion Equations and Their Linearization in Dirichlet Spaces: Dirichlet Forms, Homogeneous Spaces and Nonlinear Diffusion
Backward and Forward Linearizations of Nonlinear Diffusion
Part II. Continuity Equation and Curvature Conditions in Metric Measure Spaces: Preliminaries
Absolutely Continuous Curves in Wasserstein Spaces and Continuity Inequalities in a Metric Setting
Weighted Energy Functionals along Absolutely Continuous Curves
Dynamic Kantorovich Potentials, Continuity Equation and Dual Weighted Cheeger Energies
The $\mathrm{RCDS}^{*}(K, N)$ Condition and Its Characterizations through Weighted Convexity and Evolution Variational Inequalities
Part III. Bakry-Emery Condition and Nonlinear Diffusion: The Bakry-Emery Condition
Nonlinear Diffusion Equations and Action Estimates
The Equivalence Between $\mathrm{BE}(K, N)$ and $\mathrm{RCDS}^{*}(K, N)$
Bibliography.