Sharp upper bound for a branched transport problem coming from Ginzburg-Landau models

TL;DR

This paper establishes an upper bound on the dimension of measures in a branched transport problem related to Ginzburg-Landau models, confirming a conjecture for measures with certain regularity.

Contribution

It proves that irrigated measures with local Ahlfors regularity have dimension at most 8/5, advancing understanding of branched transport in superconductivity models.

Findings

Measures with local Ahlfors regularity have dimension ≤ 8/5.

Supports the conjecture by Conti, Serfaty, and others on measure dimension.

Connects branched transport problems to superconductivity models.

Abstract

We consider a branched transport type problem with weakly imposed boundary conditions, which can be seen as a blown-up version of a reduced model for type-I superconductors in the regime of vanishing external magnetic field. We prove that if the irrigated measure is (locally) Ahlfors regular then it is of dimension at most $8/5$ in agreement with the conjecture by Conti, the third author and Serfaty.