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
TILLI, PAOLO
2010
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.Pubblicazioni consigliate
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https://hdl.handle.net/11583/2298353
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