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Nonlinear Diffusion Equations and Curvature Conditions in Metric Measure Spaces
 
Luigi Ambrosio Scuola Normale Superiore, Pisa, Italy
Andrea Mondino University of Warwick, Coventry, United Kingdom
Giuseppe Savaré Università di Pavia, Pavia, Italy
Nonlinear Diffusion Equations and Curvature Conditions in Metric Measure Spaces
Softcover ISBN:  978-1-4704-3913-2
Product Code:  MEMO/262/1270
List Price: $81.00
MAA Member Price: $72.90
AMS Member Price: $48.60
eBook ISBN:  978-1-4704-5513-2
Product Code:  MEMO/262/1270.E
List Price: $81.00
MAA Member Price: $72.90
AMS Member Price: $48.60
Softcover ISBN:  978-1-4704-3913-2
eBook: ISBN:  978-1-4704-5513-2
Product Code:  MEMO/262/1270.B
List Price: $162.00 $121.50
MAA Member Price: $145.80 $109.35
AMS Member Price: $97.20 $72.90
Nonlinear Diffusion Equations and Curvature Conditions in Metric Measure Spaces
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Nonlinear Diffusion Equations and Curvature Conditions in Metric Measure Spaces
Luigi Ambrosio Scuola Normale Superiore, Pisa, Italy
Andrea Mondino University of Warwick, Coventry, United Kingdom
Giuseppe Savaré Università di Pavia, Pavia, Italy
Softcover ISBN:  978-1-4704-3913-2
Product Code:  MEMO/262/1270
List Price: $81.00
MAA Member Price: $72.90
AMS Member Price: $48.60
eBook ISBN:  978-1-4704-5513-2
Product Code:  MEMO/262/1270.E
List Price: $81.00
MAA Member Price: $72.90
AMS Member Price: $48.60
Softcover ISBN:  978-1-4704-3913-2
eBook ISBN:  978-1-4704-5513-2
Product Code:  MEMO/262/1270.B
List Price: $162.00 $121.50
MAA Member Price: $145.80 $109.35
AMS Member Price: $97.20 $72.90
  • Book Details
     
     
    Memoirs of the American Mathematical Society
    Volume: 2622019; 121 pp
    MSC: Primary 49; 35; Secondary 58; 31

    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.

  • Table of Contents
     
     
    • Chapters
    • 1. Introduction
    • 2. Contraction and Convexity via Hamiltonian Estimates: an Heuristic Argument
    • 1. Nonlinear Diffusion Equations and Their Linearization in Dirichlet Spaces
    • 3. Dirichlet Forms, Homogeneous Spaces and Nonlinear Diffusion
    • 4. Backward and Forward Linearizations of Nonlinear Diffusion
    • 2. Continuity Equation and Curvature Conditions in Metric Measure Spaces
    • 5. Preliminaries
    • 6. Absolutely Continuous Curves in Wasserstein Spaces and Continuity Inequalities in a Metric Setting
    • 7. Weighted Energy Functionals along Absolutely Continuous Curves
    • 8. Dynamic Kantorovich Potentials, Continuity Equation and Dual Weighted Cheeger Energies
    • 9. The $\ensuremath {\mathrm {RCD}^*(K,N)}$ Condition and Its Characterizations through Weighted Convexity and Evolution Variational Inequalities
    • 3. Bakry-Émery Condition and Nonlinear Diffusion
    • 10. The Bakry-Émery Condition
    • 11. Nonlinear Diffusion Equations and Action Estimates
    • 12. The Equivalence Between $\mathrm {BE}(K,N)$ and $\ensuremath {\mathrm {RCD}^*(K,N)}$
  • Additional Material
     
     
  • Requests
     
     
    Review Copy – for publishers of book reviews
    Permission – for use of book, eBook, or Journal content
    Accessibility – to request an alternate format of an AMS title
Volume: 2622019; 121 pp
MSC: Primary 49; 35; Secondary 58; 31

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.

  • Chapters
  • 1. Introduction
  • 2. Contraction and Convexity via Hamiltonian Estimates: an Heuristic Argument
  • 1. Nonlinear Diffusion Equations and Their Linearization in Dirichlet Spaces
  • 3. Dirichlet Forms, Homogeneous Spaces and Nonlinear Diffusion
  • 4. Backward and Forward Linearizations of Nonlinear Diffusion
  • 2. Continuity Equation and Curvature Conditions in Metric Measure Spaces
  • 5. Preliminaries
  • 6. Absolutely Continuous Curves in Wasserstein Spaces and Continuity Inequalities in a Metric Setting
  • 7. Weighted Energy Functionals along Absolutely Continuous Curves
  • 8. Dynamic Kantorovich Potentials, Continuity Equation and Dual Weighted Cheeger Energies
  • 9. The $\ensuremath {\mathrm {RCD}^*(K,N)}$ Condition and Its Characterizations through Weighted Convexity and Evolution Variational Inequalities
  • 3. Bakry-Émery Condition and Nonlinear Diffusion
  • 10. The Bakry-Émery Condition
  • 11. Nonlinear Diffusion Equations and Action Estimates
  • 12. The Equivalence Between $\mathrm {BE}(K,N)$ and $\ensuremath {\mathrm {RCD}^*(K,N)}$
Review Copy – for publishers of book reviews
Permission – for use of book, eBook, or Journal content
Accessibility – to request an alternate format of an AMS title
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