Calculus of Variations and Geometric Measure Theory

A. Figalli - N. Gigli

A new transportation distance between non-negative measures, with applications to gradients flows with Dirichlet boundary conditions

created by figalli on 10 Nov 2009
modified on 11 May 2010


Accepted Paper

Inserted: 10 nov 2009
Last Updated: 11 may 2010

Journal: J. Math. Pures Appl.
Year: 2009


In this paper we introduce a new transportation distance between non-negative measures inside a domain $\O$. This distance enjoys many nice properties, for instance it makes the space of non-negative measures inside $\O$ a geodesic space without any convexity assumption on the domain. Moreover we will show that the gradient flow of the entropy functional $\int_\O [\rho \log(\rho) - \rho]\,dx$ with respect to this distance coincides with the heat equation, subject to the Dirichlet boundary condition equal to $1$.