Calculus of Variations and Geometric Measure Theory
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R. Rios-Zertuche

Existence of differentiable curves in convex sets and the concept of direction of the flow in mass transportation

created by rios-zertuche on 06 Feb 2019



Inserted: 6 feb 2019

Year: 2018

ArXiv: 1810.05999 PDF


In this paper we consider convex subsets of locally-convex topological vector spaces. Given a fixed point in such a convex subset, we show that there exists a curve completely contained in the convex subset and leaving the point in a given direction if and only if the direction vector is contained in the sequential closure of the tangent cone at that point. We apply this result to the characterization of the existence of weakly differentiable families of probability measures on a smooth manifold and of the distributions that can arise as their derivatives. This gives us a way to consider the mass transport equation in a very general context, in which the notion of direction turns out to be given by an element of a Colombeau algebra.

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