Communications in Mathematical Sciences

Volume 20 (2022)

Number 3

Weighted irrigation plans

Pages: 611 – 651

DOI: https://dx.doi.org/10.4310/CMS.2022.v20.n3.a2

Authors

Alberto Bressan (Department of Mathematics, Pennsylvania State University, University Park, Pa., U.S.A.)

Qing Sun (Department of Mathematics, Pennsylvania State University, University Park, Pa., U.S.A.)

Abstract

We model an irrigation network where lower branches must be thicker in order to support the weight of the higher ones. This leads to a countable family of ODEs, one for each branch, that must be solved by backward induction. Having introduced conditions that guarantee the existence and uniqueness of solutions, our main result establishes the lower semicontinuity of the corresponding cost functional, w.r.t. pointwise convergence of the irrigation plans. In turn, this yields the existence of an optimal irrigation plan, in the presence of these additional weights.

Keywords

ramified transport, irrigation plan, impulsive differential equations on networks

2010 Mathematics Subject Classification

28A80, 34A37, 49Q20, 92B05

Received 7 June 2019

Received revised 16 August 2021

Accepted 23 August 2021

Published 21 March 2022