Title: An approach to topological singularities through Mumford-Shah type functionals 

Abstract:  We introduce a notion of distributional Jacobian for BV maps.  We will use such a definition 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 (GL) and core-radius (CR) approaches. 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.  We show that at any logarithmic scale our functional is “variationally equivalent” to the “standard” (CR) and (GL) model. Joint work with V. Crismale, R. Scala, and N. Van Goethem.

Title: EDP convergence for evolutionary systems with gradient flow structure

Abstract: We discuss the notion of EDP convergence for gradient systems. This convergence is based on De Giorgi's energy-dissipation principle (EDP) and allows us to study coarse-graining limits for families of gradient systems. While the energies simply converge in the sense of De Giorgi's Gamma convergence, the derivation of the effective dissipation potential is more involved. Several examples will be discussed, that show the flexibility of the theory, e.g., to allow situations where starting from quadratic dissipation potentials we arrive at effective dissipation potentials that are no longer quadratic.

Title: TBA 

Abstract: TBA