A fractal shape optimization problem in branched transport

TL;DR

This paper studies the optimal shape of unit volume sets that can be efficiently irrigated from a single source using branched transport, establishing existence, regularity, and conjecturing about their fractal boundary properties.

Contribution

It introduces a shape optimization framework for branched transport, proves existence of solutions, analyzes their regularity, and conjectures about their fractal boundary dimensions.

Findings

Existence of optimal shapes as sublevel sets of the landscape function.

Proved $eta$-H{"o}lder regularity of the landscape function.

Conjecture that the boundary has a non-integer Minkowski dimension.

Abstract

We investigate the following question: what is the set of unit volume which can be best irrigated starting from a single source at the origin, in the sense of branched transport? We may formulate this question as a shape optimization problem and prove existence of solutions, which can be considered as a sort of "unit ball" for branched transport. We establish some elementary properties of optimizers and describe these optimal sets A as sublevel sets of a so-called landscape function which is now classical in branched transport. We prove $\beta$-H{\"o}lder regularity of the landscape function, allowing us to get an upper bound on the Minkowski dimension of the boundary: dim $\partial$A $\le$ d -- $\beta$ (where $\beta$ := d($\alpha$ -- (1 -- 1/d)) $\in$ (0, 1) is a relevant exponent in branched transport, associated with the exponent $\alpha$ > 1 -- 1/d appearing in the cost). We are not able to prove the upper bound, but we conjecture that $\partial$A is of non-integer dimension d -- $\beta$. Finally, we make an attempt to compute numerically an optimal shape, using an adaptation of the phase-field approximation of branched transport introduced some years ago by Oudet and the second author.