Rectifiability of Optimal Transportation Plans

McCann, Robert J. ; Pass, Brendan ; Warren, Micah

Canadian journal of mathematics, Tome 64 (2012) no. 4, pp. 924-934

Résumé

The regularity of solutions to optimal transportation problems has become a hot topic in current research. It is well known by now that the optimal measure may not be concentrated on the graph of a continuous mapping unless both the transportation cost and the masses transported satisfy very restrictive hypotheses (including sign conditions on the mixed fourth-order derivatives of the cost function). The purpose of this note is to show that in spite of this, the optimal measure is supported on a Lipschitz manifold, provided only that the cost is ${{C}^{2}}$ with non-singular mixed second derivative. We use this result to provide a simple proof that solutions to Monge's optimal transportation problem satisfy a change of variables equation almost everywhere.

Détail

DOI : 10.4153/CJM-2011-080-6

Mots-clés : 49K20, 49K60, 35J96, 58C07

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