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

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 $\frac{1}{\epsilon }$. Here, $\epsilon$ is a length-scale parameter. We show that, in the $|\log \epsilon |$ regime, the Jacobian distributions converge, as $\epsilon \to 0^{+}$, 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 $\mu$.