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

We introduce a weak notion of $2\times 2$-minors of gradients for 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 the unit sphere in $\mathbb{R}^2$ and the energy is given by the sum of the squared $L^2$ norm of the approximate gradient $\nabla u$ 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 distributional Jacobians 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 $\mu$.