An approximation result for free discontinuity functionals by means of non-local energies

We approximate, in the sense of Gamma-convergence, free discontinuity functionals with linear growth by a sequence of non-local integral functionals depending on the average of the gradient on small balls. The result extends to a higher dimension what is already proved in (Ann. Mat. Pura Appl. 2007; 186(4): 722–744), where there is the proof of the general one-dimensional case, and in (ESAIM Control Optim. Calc. Var. 2007; 13(1):135–162), where a particular n-dimensional case is treated. Moreover, we investigate whether it is possible to approximate a given free discontinuity functional by means of non-local energies.