Journal Home Page

Cumulative Index

List of all Volumes

Complete Contents
of this Volume

Previous Article

Next Article

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


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.

[ Fulltext-pdf  (131  KB)] for subscribers only.