Γ-Convergence for the Irrigation Problem

We study the asymptotics of the functional $F(\gamma)=\int f(x) d_\gamma(x)^pdx$, where $d_\gamma$ is the distance function to $\gamma$, among all connected compact sets $\gamma$ of given length, when the prescribed length tends to infinity. After properly scaling, we prove the existence of a $\Gamma$-limit in the space of probability measures, thus retrieving information on the asymptotics of minimal sequences