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July  2005, 12(4): 607-628. doi: 10.3934/dcds.2005.12.607

Transport density in Monge-Kantorovich problems with Dirichlet conditions

Giuseppe Buttazzo 1, and  Eugene Stepanov 1,

1. 

Dipartimento di Matematica, Università di Pisa, via Buonarroti 2, 56127 Pisa, Italy

Received  January 2004 Revised  October 2004 Published  January 2005

We study the properties of the transport density measure in the Monge-Kantorovich optimal mass transport problem in the presence of so-called Dirichlet constraint, i.e. when some closed set is given along which the cost of transportation is zero. The Hausdorff dimension estimates, as well as summability and higher regularity properties of the transport density are studied. The uniqueness of the transport density is proven in the case when the masses to be transported are represented by measures absolutely continuous with respect to the Lebesgue measure.
Keywords: Monge-Kantorovich problem, regularity., Optimal transport problem, transport density.
Mathematics Subject Classification: 49Q20, 49N60, 49Q1.
Citation: Giuseppe Buttazzo, Eugene Stepanov. Transport density in Monge-Kantorovich problems with Dirichlet conditions. Discrete and Continuous Dynamical Systems, 2005, 12 (4) : 607-628. doi: 10.3934/dcds.2005.12.607
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