Calculus of Variations and Geometric Measure Theory

L. Caravenna

An existence result for the Monge problem in $R^n$ with norm cost functions

created by caravenna on 06 Jun 2009
modified on 10 Sep 2009


Inserted: 6 jun 2009
Last Updated: 10 sep 2009

Year: 2009


We establish existence of solutions to the Monge problem in $R^n$ with a norm cost function, assuming absolute continuity of the initial measure.

The proof is based on a one dimensional reduction argument.

The loss in strict convexity implies that transport is possible along several directions. As for crystalline norms, we single out particular solutions to the Kantorovich relaxation with a secondary variational problem involving a strictly convex norm. We then define a map rearranging the mass within the rays, with a one-dimensional Sudakov-type argument proved by a regularity of the disintegration of the Lebesgue measure w.r.t. the rays.

Remark: The construction presently given in the preprint needs the further technical assumption that, with the notation of Section 3.4, the set of points $x$ in $\mathcal{T}_{\mathrm s}$ whose secondary transport ray belongs to the relative border of the convex envelope of $\{y:\phi(x)-\phi(y)=| y-x|\}$ is $\mu$-negligible.\end{abstract}