# Interfaces and Free Boundaries

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**Volume 11, Issue 2, 2009, pp. 201–238**

**DOI: 10.4171/IFB/209**

Partial *L*^{1} Monge–Kantorovich problem: variational formulation and numerical approximation

John W. Barrett (1) and Leonid Prigozhin (2)

(1) Department of Mathematics, Imperial College London, South Kensington Campus, SW7 2AZ, LONDON, UNITED KINGDOM(2) J. Blaustein Institute for Desert Research, Ben Gurion University of the Negev, Sede Boqer Campus, 84990, BEER-SHEBA, ISRAEL

We consider the Monge–Kantorovich problem with transportation cost equal to distance and a relaxed mass balance condition: instead of optimally transporting one given distribution of mass onto another with the same total mass, only a given amount of mass, m, has to be optimally transported. In this partial problem the given distributions are allowed to have different total masses and m should not exceed the least of them. We derive and analyze a variational formulation of the arising free boundary problem in optimal transportation. Furthermore, we introduce and analyse the ﬁnite element approximation of this formulation using the lowest order Raviart–Thomas element. Finally, we present some numerical experiments where both approximations to the optimal transportation domains and the optimal transport between them are computed.

*Keywords: *Monge–Kantorovich problem, optimal transportation, free boundary, variational formulation, ﬁnite elements, augmented Lagrangian, convergence analysis