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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| Hlavní autoři: | , , |
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| Médium: | E-kniha Kniha |
| Jazyk: | angličtina |
| Vydáno: |
Providence, Rhode Island
American Mathematical Society
2019
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| Vydání: | 1 |
| Edice: | Memoirs of the American Mathematical Society |
| Témata: | |
| ISBN: | 9781470439132, 1470439131 |
| ISSN: | 0065-9266, 1947-6221 |
| On-line přístup: | Získat plný text |
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- Introduction -- Contraction and Convexity via Hamiltonian Estimates: an Heuristic Argument -- Nonlinear Diffusion Equations and Their Linearization in Dirichlet Spaces -- Dirichlet Forms, Homogeneous Spaces and Nonlinear Diffusion -- Backward and Forward Linearizations of Nonlinear Diffusion -- 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 <inline-formula content-type="math/mathml"> R C D ∗ ( K , N ) \mathrm {RCD}^*(K,N) </inline-formula> Condition and Its Characterizations through Weighted Convexity and Evolution Variational Inequalities -- Bakry-Émery Condition and Nonlinear Diffusion -- The Bakry-Émery Condition -- Nonlinear Diffusion Equations and Action Estimates -- The Equivalence Between <inline-formula content-type="math/mathml"> B E ( K , N ) \mathrm {BE}(K,N) </inline-formula> and <inline-formula content-type="math/mathml"> R C D ∗ ( K , N ) \mathrm {RCD}^*(K,N) </inline-formula>
- Cover -- Title page -- Chapter 1. Introduction -- Chapter 2. Contraction and Convexity via Hamiltonian Estimates: an Heuristic Argument -- Part I . Nonlinear Diffusion Equations and Their Linearization in Dirichlet Spaces -- Chapter 3. Dirichlet Forms, Homogeneous Spaces and Nonlinear Diffusion -- 3.1. Dirichlet forms -- 3.2. Completion of quotient spaces w.r.t. a seminorm -- 3.3. Nonlinear diffusion -- Chapter 4. Backward and Forward Linearizations of Nonlinear Diffusion -- Part II . Continuity Equation and Curvature Conditions in Metric Measure Spaces -- Chapter 5. Preliminaries -- 5.1. Absolutely continuous curves, Lipschitz functions and slopes -- 5.2. The Hopf-Lax evolution formula -- 5.3. Measures, couplings, Wasserstein distance -- 5.4. _{ }-absolutely continuous curves and dynamic plans -- 5.5. Metric measure spaces and the Cheeger energy -- 5.6. Entropy estimates of the quadratic moment and of the Fisher information along nonlinear diffusion equations -- 5.7. Weighted Γ-calculus -- Chapter 6. Absolutely Continuous Curves in Wasserstein Spaces and Continuity Inequalities in a Metric Setting -- Chapter 7. Weighted Energy Functionals along Absolutely Continuous Curves -- Chapter 8. Dynamic Kantorovich Potentials, Continuity Equation and Dual Weighted Cheeger Energies -- Chapter 9. The \RCDS Condition and Its Characterizations through Weighted Convexity and Evolution Variational Inequalities -- 9.1. Green functions on intervals -- 9.2. Entropies and their regularizations -- 9.3. The \CDS condition and its characterization via weighted action convexity -- 9.4. \RCD ∞ spaces and a criterium for \CDS via -- Part III . Bakry-Émery Condition and Nonlinear Diffusion -- Chapter 10. The Bakry-Émery Condition -- 10.1. The Bakry-Émery condition for local Dirichlet forms and interpolation estimates
- 10.2. Local and "nonlinear" characterization of the metric \BE condition in locally compact spaces -- Chapter 11. Nonlinear Diffusion Equations and Action Estimates -- Chapter 12. The Equivalence Between \BE and \RCDS -- 12.1. Regular curves and regularized entropies -- 12.2. \BE yields for regular entropy functionals in \MC -- 12.3. \RCDS implies \BE -- Bibliography -- Back Cover