An extension theorem for special functions with bounded variation and an application to homogenization of Mumford-Shah type energies

The result I will present is the existence of an extension operator for special functions with bounded variation with a careful energy estimate.

The main application is a compactness result for non-coercive functionals consisting of a volume and a surface integral, as those occurring in computer vision and in the mathematical theory of elasticity for brittle materials. More precisely I will focus on the study of the asymptotic behaviour of the Mumford-Shah functional on a periodically perforated domain, as the size of the holes and the periodicity parameter of the structure tend to zero.