Transport density in Monge-Kantorovich problems with Dirichlet conditions

Giuseppe Buttazzo; Eugene Stepanov

doi:10.3934/dcds.2005.12.607

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2005, Volume 12, Issue 4: 607-628. Doi: 10.3934/dcds.2005.12.607

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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

Received: January 2004

Revised: October 2004

Published: July 2005

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Abstract

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.

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Mathematics Subject Classification: 49Q20, 49N60, 49Q10.

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