# American Institute of Mathematical Sciences

September  2021, 16(3): 493-511. doi: 10.3934/nhm.2021014

## Irrigable measures for weighted irrigation plans

 Department of Mathematics, Penn State University, University Park, PA 16803, USA

Received  January 2020 Revised  April 2021 Published  September 2021 Early access  July 2021

A model of irrigation network, where lower branches must be thicker in order to support the weight of the higher ones, was recently introduced in [7]. This leads to a countable family of ODEs, describing the thickness of every branch, solved by backward induction. The present paper determines what kind of measures can be irrigated with a finite weighted cost. Indeed, the boundedness of the cost depends on the dimension of the support of the irrigated measure, and also on the asymptotic properties of the ODE which determines the thickness of branches.

Citation: Qing Sun. Irrigable measures for weighted irrigation plans. Networks & Heterogeneous Media, 2021, 16 (3) : 493-511. doi: 10.3934/nhm.2021014
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Left: A free standing tree with 5 branches. In this example, $\mathcal{O}(1) = \{2,3 \},\mathcal{O}(3) = \{4,5\}, \mathcal{O}(2) = \mathcal{O}(4) = \mathcal{O}(5) = \emptyset$. Right: On each branch, the weight decreases as one moves from the lower portion to the tip
Left: Two finite truncation plans, showing three maximal $\varepsilon$-good paths (thick lines) and six maximal $\varepsilon'$-good paths (thin lines), for $0<\varepsilon'<\varepsilon$. Right: The three maximal $\varepsilon$-good paths can be partitioned into five elementary branches, by the Path Splitting Algorithm
Left: The dyadic approxmiated measure $\mu_1$ is supported on the four centers $x_1^1,\ldots,x^1_4$ of small cubes. Right: Dyadic approximated measures corresponding to a family of partitions into dyadic cubes in $\bf{R}^2$
The dyadic irrigation plans in $\bf{R}^2$. Left: The dyadic irrigation plan $\chi_1$. The multiplicity on each branch equals to the mass on the terminal point. Right: The dyadic irrigation plan $\chi_2$. The particles are first transported to the 4 centers in $\mathcal{P}_1$, then on each center in $\mathcal{P}_1$, the particles are transported to the neighboring 4 centers in $\mathcal{P}_2$
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