Γ-Convergence for the Irrigation Problem

We study the asymptotics of the functional

, where

d_\gamma d γ

is the distance function to

\gamma γ

, among all connected compact sets

\gamma γ

of given length, when the prescribed length tends to infinity. After properly scaling, we prove the existence of a

\Gamma Γ

-limit in the space of probability measures, thus retrieving information on the asymptotics of minimal sequences