Calculus of Variations and Geometric Measure Theory
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P. Pegon - F. Santambrogio - Q. Xia

A fractal shape optimization problem in branched transport

created by santambro on 01 Sep 2017

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Submitted Paper

Inserted: 1 sep 2017
Last Updated: 1 sep 2017

Year: 2017

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 $\beta$-H\"older 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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