A fractal shape optimization problem in branched transport
Résumé
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 β-Hölder regularity of the landscape function, allowing us to get an upper bound on the Minkowski dimension of the boundary: dim ∂A ≤ d − β (where β := d(α − (1 − 1/d)) ∈ (0, 1) is a relevant exponent in branched transport, associated with the exponent α > 1 − 1/d appearing in the cost). We are not able to prove the upper bound, but we conjecture that ∂A is of non-integer dimension d − β. 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.
Origine : Fichiers produits par l'(les) auteur(s)
Loading...