Nonlinear diffusion equations and curvature conditions in metric measure spaces

著者

• Ambrosio, Luigi
• Mondino, Andrea
• Savare, Giuseppe

書誌事項

Nonlinear diffusion equations and curvature conditions in metric measure spaces

Luigi Ambrosio, Andrea Mondino, Giuseppe Savare

（Memoirs of the American Mathematical Society, no. 1270）

American Mathematical Society, c2019

注記

"November 2019, volume 262, number 1270 (seventh of 7 numbers)"

Includes bibliographical reference (p. 119-121)

内容説明・目次

内容説明

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.

目次

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.

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詳細情報

• NII書誌ID(NCID): BB29722493
• ISBN:
  • 9781470439132
• 出版国コード: us
• タイトル言語コード: eng
• 本文言語コード: eng
• 出版地: Providence, R.I.
• ページ数/冊数: v, 121 p.
• 大きさ: 26 cm
• 親書誌ID:
  • BA00035927