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New dimensional bounds for a branched transport problem

Alessandro Cosenza, Michael Goldman, Melanie Koser

TL;DR

The paper analyzes a reduced branched transport model with a Sobolev boundary penalty to capture boundary patterning in type-I superconductors. It develops a framework linking local energy scaling to fractal dimensions of the irrigated boundary measure, leveraging McCann interpolation, the no-loop property in 1D, and Ahlfors regularity assumptions. The main results show that for s \\leq 1/4 the boundary measure has lower dimension 1, while for s \\gt 1/4 the dimension is pinned by ᾱ depending on regularity: dim_{F} \\mu_T \\leq ᾱ and dim_M \\mu_T \\geq ᾱ, with sharp conclusions when mu_T is upper or lower \\alpha-Ahlfors regular; in particular, mu_T cannot be smooth. By combining local energy bounds with a first variation argument, the authors prove a rigorous fractal-type lower bound and identify regimes where the boundary exhibits non-integer dimensionality, offering a rigorous justification for fractal boundary behavior in this branched transport reduction.

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.

New dimensional bounds for a branched transport problem

TL;DR

The paper analyzes a reduced branched transport model with a Sobolev boundary penalty to capture boundary patterning in type-I superconductors. It develops a framework linking local energy scaling to fractal dimensions of the irrigated boundary measure, leveraging McCann interpolation, the no-loop property in 1D, and Ahlfors regularity assumptions. The main results show that for s \\leq 1/4 the boundary measure has lower dimension 1, while for s \\gt 1/4 the dimension is pinned by ᾱ depending on regularity: dim_{F} \\mu_T \\leq ᾱ and dim_M \\mu_T \\geq ᾱ, with sharp conclusions when mu_T is upper or lower \\alpha-Ahlfors regular; in particular, mu_T cannot be smooth. By combining local energy bounds with a first variation argument, the authors prove a rigorous fractal-type lower bound and identify regimes where the boundary exhibits non-integer dimensionality, offering a rigorous justification for fractal boundary behavior in this branched transport reduction.

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.

Paper Structure

This paper contains 17 sections, 28 theorems, 214 equations, 5 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Notation
  4. Optimal transport
  5. The internal energy functional
  6. The case $d=1$
  7. Negative Sobolev spaces
  8. The main functional and global scaling laws
  9. From local scaling laws to bounds on the lower and upper dimension of $\mu_T$
  10. Dimension of a measure
  11. Local scaling laws and dimensional estimates
  12. Local scaling laws for minimizers with upper $\alpha$-Ahlfors regular boundary measures
  13. Interpolated measures, upper $\alpha$-Ahlfors regularity and Sobolev regularity
  14. Local scaling laws for minimizers with upper $\alpha$-Ahlfors regular boundary measure
  15. Local scaling laws for minimizers with lower $\alpha$-Ahlfors regular boundary measures
  16. ...and 2 more sections

Key Result

Theorem 1.1

Let $d=1$, $s\in (0,1)$ and $\mu$ be a minimizer of $E_{s,T}$. If $s\le 1/4$ then (see Section sec:dim for precise definitions) $\underline{\textup{dim}}\,\mu_T=1$. If instead $s>\frac{1}{4}$ and $\mu_T$ is upper $\alpha$-Ahlfors regular, then whereas if $\mu_T$ is lower $\alpha$-Ahlfors regular, then

Figures (5)

  • Figure 1: Representation of the typical case for the $3$ intervals $J^1(t)$, $J^2(t)$ and $J^3(t)$ for $t\in [0,T]$.
  • Figure 2: On the left a sketch of $\mu$ on $[T-\varepsilon,T]$, on the right a sketch of $\nu$ on $[T-\varepsilon,T]$.
  • Figure 3: Illustration of the construction on one single branch for $d=1$. The support of $\mu_t$ is contained in the blue domain.
  • Figure 4: Sketch of the described decomposition with respect to $J_i$. The part of the minimizer which irrigates $J_i$ is contained in the blue domain.
  • Figure 5: Sketch of the local construction within the cones.

Theorems & Definitions (58)

  • Theorem 1.1
  • Remark 1.2
  • Remark 1.3
  • Remark 1.4
  • Theorem 1.5
  • Theorem 1.6
  • Definition 2.1
  • Proposition 2.2: Subsystems
  • Lemma 2.3: No-loop condition
  • Proposition 2.4
  • ...and 48 more