Calculus of Variations and Geometric Measure Theory

E. Paolini - E. Stepanov

Connecting measures by means of branched transportation networks at finite cost

created by stepanov on 11 Nov 2006
modified on 20 Feb 2009


Published Paper

Inserted: 11 nov 2006
Last Updated: 20 feb 2009

Journal: J. Math. Sciences (N.Y.)
Volume: 157
Number: 6
Pages: 858-873
Year: 2009


The paper studies the couples of finite Borel measures $\varphi_0$ and $\varphi_1$ with compact supports in $*R*^n$ which can be transported to each other at a finite $W^\alpha$ cost, where \[ W^\alpha(\varphi_0,\varphi_1):=\inf \{*M*^\alpha(T)\,:\, \partial T= \varphi_0-\varphi_1\}, \qquad\alpha\in [0,1], \] the infimum being taken over real normal currents of finite mass, and $*M*^\alpha(T)$ standing for the $\alpha$-mass of $T$. Besides the class of $\alpha$-irrigable measures (i.e.\ measures which can be transported to a Dirac measure with the appropriate total mass at a finite $W^\alpha$ cost), two other important classes of measures are studied, which are called in the paper purely $\alpha$-non-irrigable and marginally $\alpha$-non-irrigable, and are in a certain sense complementary to each other. For instance, purely $\alpha$-non-irrigable and Ahlfors-regular measures are, roughly speaking, those having sufficiently high dimension. One shows that for $\varphi_0$ to be transported to $\varphi_1$ at finite $W^\alpha$ cost their naturally defined purely $\alpha$-non-irrigable parts have to coincide.

Keywords: optimal transportation, Branched transport, transportation network, irrigability, irrigation problem