Nonlinear Diffusion Equations and Curvature Conditions in Metric Measure Spaces
By Luigi Ambrosio, Andrea Mondino, Giuseppe Savar'e. Memoirs of the American Mathematical Society, 2015
187 citations recorded by the source corpus.
Aim of this paper is to provide new characterizations of the curvature dimension condition in the context of metric measure spaces (X,d,m). On the geometric side, our 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, our 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 CD*(K,N) condition of Bacher-Sturm.
Subfields: Applied Mathematics
View Paper: https://www.semanticscholar.org/paper/b38807ed05f5e7bd4a784a2a105db154550cd13b
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In Applied Mathematics
- Elia Brué
- Andrea Mondino
- Enrico Pasqualetto
- Semola Daniele
- Daniele Semola
- A. Naber
- Michel Ledoux
- Nicola Gigli
Citations in the corpus
- On Optimal Matching of Gaussian Samples (2019, background): “More general frameworks covering these instances are the settings of Markov triples (E, μ,Γ) of [6] and of RCD∗(K,N) Riemannian metric measure spaces studied in [2, 3, 18] to which most of the conclusions developed here may be transferred.”
- Constancy of the Dimension for RCD(K,N) Spaces via Regularity of Lagrangian Flows (2018, background): “was proposed in [38] (see also [12, 33]) as a finite-dimensional counterpart to RCD.”
- Rectifiability of the reduced boundary for sets of finite perimeter over $\RCD(K,N)$ spaces. (2019, methodology): “We refer to [10, 25] for a more detailed discussion and equivalent characterizations of the RCD(K,N) condition.”
- Structure theory of metric measure spaces with lower Ricci curvature bounds (2014, background): “While a great deal of progress has been made in this latter general framework, see for instance [3, 5, 6, 7, 8, 9, 10, 14, 15, 27, 28, 29, 30, 32, 33, 34, 40, 41, 42, 46], the structure theory on such metric-measure spaces is still much less developed than in the case of smooth limits.”