2017/09/05 by Paul Pegon, Filippo Santambrogio, Pegon, Paul +3 · 1 citation
Computer Science · Mathematics · Physics and Astronomy · #Advanced Mathematical Modeling in Engineering #Classical Analysis and ODEs (math.CA) #FOS: Mathematics #Geometric Analysis and Curvature Flows #Optimization and Control (math.OC) #Theoretical and Computational Physics
paper · pdf · doi:10.48550/arxiv.1709.01415
openalex publication_date 2017/09/05 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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.