| Abstract: |
| We will present variational approaches to the analysis of topological singularities in the plane, starting from the - nowadays - classical Ginzburg-Landau (GL) model and core-radius (CR) approach. We will introduce a third approach inspired by the Mumford-Shah functional used in the context of image segmentation. Within our framework, the order parameter is an $SBV$ map taking values in the unit sphere of the plane; the bulk energy is the squared $L^2$ norm of the approximate gradient whereas the penalization term is given by the length of the jump set, scaled by a small parameter. After providing a notion of Jacobian determinant for $SBV$ maps, we show that at any logarithmic scale our functional is ``variationally equivalent'' to the ``standard'' (CR) and (GL) models. Joint work with Vito Crismale, Nicolas Van Goethem and Riccardo Scala. |
|