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

摘要

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