New dimensional bounds for a branched transport problem

TL;DR

This paper proves in a simplified 2D setting that the boundary measure in a branched transport problem exhibits non-integer dimension, confirming a conjecture related to pattern formation in superconductors.

Contribution

It provides the first rigorous proof of singular boundary measure behavior in a minimality-driven branched transport model under Ahlfors regularity.

Findings

Boundary measure has non-integer dimension.

First rigorous proof of singular behavior in irrigated measures.

Supports conjecture on pattern formation in superconductors.

Abstract

We consider a branched transport problem with weakly imposed boundary conditions. This problem arises as a reduced model for pattern formation in type-I superconductors. For this model, it is conjectured that the dimension of the boundary measure is non-integer. We prove this conjecture in a simplified 2D setting, under the (strong) assumption of Ahlfors regularity of the irrigated measure. This work is the first rigorous proof of a singular behaviour for irrigated measures resulting from minimality.