2022/05/29 by De Luca, Lucia, Scala, Riccardo, Van Goethem, Nicolas
#Analysis of PDEs (math.AP) #FOS: Mathematics
paper · doi:10.48550/arxiv.2205.14746
We introduce a weak notion of 2× 2-minors of gradients of 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 \mathbbS1 and the energy is made by the sum of the squared L2 norm of ∇ u and of the length of (the closure of) the jump set of u multiplied by \frac 1 ε. Here, ε is a length-scale parameter. We show that, in the |logε| regime, the Jacobian distributions 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.