*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

**Abstract:**

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

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