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.

Some explicit examples of minimizers for the irrigation problem / Tilli, Paolo. - In: JOURNAL OF CONVEX ANALYSIS. - ISSN 0944-6532. - STAMPA. - 17:2(2010), pp. 583-595.

### Some explicit examples of minimizers for the irrigation problem

#### Abstract

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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2010
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Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/11583/2298353
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