Finite Difference Approximation of Free Discontinuity Problems

Abstract

We approximate functionals depending on the gradient of $u$ and on the behaviour of $u$ near the discontinuity points, by families of non-local functionals where the gradient is replaced by finite differences. We prove pointwise convergence, $Γ$-convergence, and a compactness result which implies, in particular, the convergence of minima and minimizers.