*Published Paper*

**Inserted:** 1 sep 2017

**Last Updated:** 9 apr 2024

**Journal:** Journal de Mathématiques Pures et Appliquées

**Volume:** 123

**Pages:** 244-269

**Year:** 2019

**Doi:** 10.1016/j.matpur.2018.06.007

**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 \enquote{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 $\beta$-Hölder regularity of the landscape function, allowing us to get an upper bound on the Minkowski dimension of the boundary: $\overline{\dim}_M \partial A \leq d-\beta$ (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.

**Keywords:**
Branched transport, landscape function, Morrey-Campanato spaces, phase-field approximation, non-smooth optimization

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