Some Explicit Examples of Minimizers for the Irrigation Problem

We construct some examples of explicit solutions to the problem

where the minimum is over all connected compact sets

\gamma\subset \overline\Omega\subset{\mathbb R}^2 γ ⊂ Ω ‾ ⊂ R 2

of prescribed one-dimensional Hausdorff measure. More precisely we show that, if

\gamma γ

is a

C^{1,1} C 1 , 1

curve of length

l l

with curvature bounded by

1/R 1 / R

,

l \leq\pi R l ≤ π R

and

\varepsilon\leq R ε ≤ R

, then

\gamma γ

is a solution to the above problem with

\Omega Ω

being the

\varepsilon ε

-neighbourhood of

\gamma γ

. In particular,

C^{1,1} C 1 , 1

regularity is optimal for this problem