Geodesic
Some Explicit Examples of Minimizers for the Irrigation Problem
Journal of convex analysis, Tome 17 (2010) no. 2, pp. 583-595 
P. Tilli. Some Explicit Examples of Minimizers for the Irrigation Problem. Journal of convex analysis, Tome 17 (2010) no. 2, pp. 583-595. http://geodesic.mathdoc.fr/item/JCA_2010_17_2_JCA_2010_17_2_a13/
@article{JCA_2010_17_2_JCA_2010_17_2_a13,
     author = {P. Tilli},
     title = {Some {Explicit} {Examples} of {Minimizers} for the {Irrigation} {Problem}},
     journal = {Journal of convex analysis},
     pages = {583--595},
     year = {2010},
     volume = {17},
     number = {2},
     url = {http://geodesic.mathdoc.fr/item/JCA_2010_17_2_JCA_2010_17_2_a13/}
}
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Voir la notice de l'article provenant de la source Heldermann Verlag

We construct some examples of explicit solutions to the problem \[ \min_\gamma \int_\Omega d_\gamma(x)\,dx \] where the minimum is over all connected compact sets $\gamma\subset \overline\Omega\subset{\mathbb R}^2$ of prescribed one-dimensional Hausdorff measure. More precisely we show that, if $\gamma$ is a $C^{1,1}$ curve of length $l$ with curvature bounded by $1/R$, $l \leq\pi R$ and $\varepsilon\leq R$, then $\gamma$ is a solution to the above problem with $\Omega$ being the $\varepsilon$-neighbourhood of $\gamma$. In particular, $C^{1,1}$ regularity is optimal for this problem.