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

We introduce a weak notion of 2 × 2-minors of gradients for a suitable subclass of BV functions. In the case of maps in BV(R2; R2) such a notion extends the standard definition of Jacobian determinant to non-Sobolev maps. We use this distributional Jacobian to prove a compactness and Γ -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 R2 and the energy is given by the sum of the squared L2 norm of the approximate gradient ∇u and of the length of (the closure of) the jump set of u multiplied by 1/ε. Here, ε is a length-scale parameter. We show that, in the | log ε| regime, the distributional Jacobians converge, as ε → 0+, to a finite sum µ of Dirac deltas with weights multiple of π, and that the corresponding effective energy is given by the total variation of µ.