Calculus of Variations and Geometric Measure Theory

J. M. Mazón - J. Rossi - J. Toledo

AN OPTIMAL TRANSPORTATION PROBLEM WITH A COST GIVEN BY THE EUCLIDEAN DISTANCE PLUS IMPORT/EXPORT TAXES ON THE BOUNDARY

created by mazón on 12 Jul 2012
modified on 16 Jul 2012

[BibTeX]

Submitted Paper

Inserted: 12 jul 2012
Last Updated: 16 jul 2012

Year: 2012

Abstract:

In this paper we analyze a mass transportation problem in a bounded domain in which there is the possibility of importexport mass across the boundary paying a tax fee in addition to the transport cost that is assumed to be given by the Euclidean distance. We show a general duality argument and for the dual problem we find a Kantorovich potential as the limit as $p\to \infty$ of solutions to $p-$Laplacian type problems with non linear boundary conditions. In addition, we show that this limit encodes all the relevant information for our problem, it provides the masses that are exported and imported from the boundary and also allows us the construction of an optimal transport plan. Finally we show that the arguments can be adapted to deal with the case in which the amount of mass that can be exportedimported is bounded by prescribed functions.


Download: