Calculus of Variations and Geometric Measure Theory

G. Buttazzo - E. Stepanov

Transport density in Monge-Kantorovich problems with Dirichlet conditions

created on 24 Jul 2001
modified by stepanov on 15 Oct 2006


Published Paper

Inserted: 24 jul 2001
Last Updated: 15 oct 2006

Journal: Discrete Contin. Dyn. Syst.
Volume: 12
Number: 4
Pages: 607-628
Year: 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: Optimal transport, regularity, Transport density, Monge-Kantorovich problem