Calculus of Variations and Geometric Measure Theory

L. Ambrosio - A. Mondino - G. Savaré

Nonlinear diffusion equations and curvature conditions in metric measure spaces

created by ambrosio on 21 Sep 2015
modified by mondino on 23 Mar 2017


Accepted Paper

Inserted: 21 sep 2015
Last Updated: 23 mar 2017

Journal: Memoirs Amer. Math. Soc.
Year: 2015


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.