The Monge problem in \(\mathbb R^d\). (English) Zbl 1232.49050
Summary: We first consider the Monge problem in a convex bounded subset of \({\mathbb R}^d\). The cost is given by a general norm, and we prove the existence of an optimal transport map under the classical assumption that the first marginal is absolutely continuous with respect to the Lebesgue measure. In the final part of the paper we show how to extend this existence result to a general open subset of \({\mathbb R}^d\).
MSC:
| 49Q20 | Variational problems in a geometric measure-theoretic setting |
| 49J45 | Methods involving semicontinuity and convergence; relaxation |
| 49K30 | Optimality conditions for solutions belonging to restricted classes (Lipschitz controls, bang-bang controls, etc.) |
| 93E20 | Optimal stochastic control |
Keywords:
Monge problem; existence of an optimal transport map; absolute continuity of the first marginalReferences:
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