A new approach to topological singularities via a weak notion of Jacobian for functions of bounded variation

TL;DR

This paper develops a weak Jacobian notion for BV functions to analyze topological singularities, proving compactness and Gamma-convergence results for a model similar to Ginzburg-Landau, with applications to vortex-like structures.

Contribution

It introduces a novel distributional Jacobian for BV maps, extending classical concepts to non-Sobolev functions, and applies it to a new variational model for topological singularities.

Findings

Jacobian distributions converge to sums of Dirac deltas as epsilon approaches zero

Effective energy is given by the total variation of the limiting measure

Results extend analysis of singularities in BV maps with applications to vortex models

Abstract

We introduce a weak notion of $2\times 2$-minors of gradients of a suitable subclass of $BV$ functions. In the case of maps in $BV(\mathbb{R}^2;\mathbb{R}^2)$ such a notion extends the standard definition of Jacobian determinant to non-Sobolev maps. We use this distributional Jacobian to prove a compactness and $\Gamma$-convergence result for a new model describing the emergence of topological singularities in two dimensions, in the spirit of Ginzburg-Landau and core-radius approaches. Within our framework, the order parameter is an $SBV$ map $u$ taking values in $\mathbb{S}^1$ and the energy is made by the sum of the squared $L^2$ norm of $\nabla u$ and of the length of (the closure of) the jump set of $u$ multiplied by $\frac 1 \varepsilon$. Here, $\varepsilon$ is a length-scale parameter. We show that, in the $|\log\varepsilon|$ regime, the Jacobian distributions converge, as $\varepsilon\to 0^+$, to a finite sum $\mu$ of Dirac deltas with weights multiple of $\pi$, and that the corresponding effective energy is given by the total variation of $\mu$.