Abstract

We construct some examples of explicit solutions to the problem min⁡γ∫Ωdγ(x) dx\min_\gamma \int_\Omega d_\gamma(x)\,dx where the minimum is over all connected compact sets γ⊂Ω‾⊂R2\gamma\subset \overline\Omega\subset{\mathbb R}^2 of prescribed one-dimensional Hausdorff measure. More precisely we show that, if γ\gamma is a C1,1C^{1,1} curve of length ll with curvature bounded by 1/R1/R, l≤πRl \leq\pi R and ε≤R\varepsilon\leq R, then γ\gamma is a solution to the above problem with Ω\Omega being the ε\varepsilon-neighbourhood of γ\gamma. In particular, C1,1C^{1,1} regularity is optimal for this problem

Author information

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Paolo Tilli

Dip. di Matematica, Politecnico di Torino, Corso Duca degli Abruzzi 24, 10129 Torino, Italy

paolo.tilli@polito.it

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Suggested citation

P. Tilli. “Some Explicit Examples of Minimizers for the Irrigation Problem.” Journal of Convex Analysis 17 (2010), No. 2, 583–595. https://doi.org/10.68381/jca17039

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Published by Heldermann Verlag, 2010. Rights now held by Banach Press.