Calculus and heat flow in metric measure spaces and applications to spaces with Ricci bounds from below

• 成果类型: Article
• 署名作者: Ambrosio, Luigi; Gigli, Nicola; Savare, Giuseppe
• 署名单位: Scuola Normale Superiore di Pisa; Universite Cote d'Azur; University of Pavia
• 刊物名称: INVENTIONES MATHEMATICAE
• ISSN/ISSBN: 0020-9910
• DOI: 10.1007/s00222-013-0456-1
• 发表日期: 2014
• 页码: 289-391
• 关键词: curvature conditions; EXISTENCE; EQUATIONS; geometry

摘要

This paper is devoted to a deeper understanding of the heat flow and to the refinement of calculus tools on metric measure spaces (X, d, m). Our main results are: A general study of the relations between the Hopf-Lax semigroup and Hamilton-Jacobi equation in metric spaces (X, d). The equivalence of the heat flow in L-2(X, m) generated by a suitable Dirichlet energy and the Wasserstein gradient flow of the relative entropy functional Entm in the space of probability measuresP(X). The proof of density in energy of Lipschitz functions in the Sobolev space W-1,W-2(X, d, m). A fine and very general analysis of the differentiability properties of a large class of Kantorovich potentials, in connection with the optimal transport problem, is the fourth achievement of the paper. Our results apply in particular to spaces satisfying Ricci curvature bounds in the sense of Lott and Villani (Ann. Math. 169: 903-991, 2009) and Sturm (Acta Math. 196: 65-131, 2006, and Acta Math. 196: 133-177, 2006) and require neither the doubling property nor the validity of the local Poincare inequality.