Calculus and heat flow in metric measure spaces and applications to spaces with Ricci bounds from below
Calculus and heat flow in metric measure spaces and applications to spaces with Ricci bounds from below
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DOI：
10.1007/s00222-013-0456-1
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发表时间：
2014-02-01
影响因子：
3.1
通讯作者：
Savare, Giuseppe
中科院分区：
文献类型：
--
作者：
Ambrosio, Luigi;Gigli, Nicola;Savare, Giuseppe
本文致力于对热流的深入理解和对度量度量空间（X, d, m）上的微积分工具的改进。我们的主要成果是：测度空间中Hopf-Lax半群与Hamilton-Jacobi方程关系的一般研究（X, d）。适当的狄利克雷能量在L-2（X, m）中产生的热流与相对熵泛函Entm在概率测度空间p (X)中的Wasserstein梯度流的等价性。Sobolev空间W-1，W-2（X, d, m）中Lipschitz函数能量密度的证明。本文的第四个成果，是对与最优输运问题有关的一大类Kantorovich势的可微性进行了精细而非常一般的分析。我们的结果特别适用于在Lott和Villani （Ann）的意义上满足里奇曲率界的空间。数学，169:903-991,2009)和Sturm（数学学报，196:65-131,2006，和数学学报，196:133-177,2006），并且既不需要倍增性质也不需要局部庞加莱不等式的有效性。
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.