Nonlinear Diffusion Equations and Curvature Conditions in Metric Measure Spaces.

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.

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.

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Electronic reproduction. Ann Arbor, Michigan : ProQuest Ebook Central, 2024. Available via World Wide Web. Access may be limited to ProQuest Ebook Central affiliated libraries.