Journal of Convex Analysis 17 (2010), No. 2, 583--595
Copyright Heldermann Verlag 2010
Some Explicit Examples of Minimizers for the Irrigation Problem
Paolo Tilli
Dip. di Matematica, Politecnico di Torino, Corso Duca degli Abruzzi 24, 10129 Torino, Italy
paolo.tilli@polito.it
[Abstract-pdf]
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.
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