Calculus of Variations and Geometric Measure Theory

G. Buttazzo - E. Oudet - E. Stepanov

Optimal transportation problems with free Dirichlet regions

created on 10 Dec 2001
modified on 02 Dec 2017


Published Paper

Inserted: 10 dec 2001
Last Updated: 2 dec 2017

Journal: Progress in Nonlinear Diff. Equations and their Applications
Volume: 51
Pages: 41-65
Year: 2002


A Dirichlet region for an optimal mass transportation problem is, roughly speaking, a zone in which the transportation cost is vanishing. We study the optimal transportation problem with an unknown Dirichlet region $\Sigma$ which varies in the class of closed connected subsets having prescribed $1$-dimensional Hausdorff measure. We show the existence of an optimal $\Sigma_{opt}$ and study some of its geometrical properties. We also present numerical computations which show the shape of $\Sigma_{opt}$ in some model examples.

Keywords: Monge-Kantorovich problem, mass transport problem, free boundary problem, urban planning problem