A non-local approximation of free discontinuity problems in SBV and SBD

We consider a class of functionals which are defined in the spaces SBV and SBD and which do not depend on the traces of u on the set of discontinuity points. In this work we prove that it is possible to approximate these energies, in the sense of Gamma-convergence, by means of a family of non-local functionals defined in Sobolev spaces. Moreover we illustrate some applications for image processing and mechanics.