A fractal shape optimization problem in branched transport

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 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.
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Submitted on : Thursday, December 28, 2017 - 12:08:40 PM
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  • HAL Id : hal-01581675, version 2
  • ARXIV : 1709.01415

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Paul Pegon, Filippo Santambrogio, Qinglan Xia. A fractal shape optimization problem in branched transport. Journal de Mathématiques Pures et Appliquées, Elsevier, 2019. ⟨hal-01581675v2⟩

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