Finite-difference approximation of free-discontinuity problems
Massimo Gobbino, Maria Giovanna Mora · Proceedings of the Royal Society of Edinburgh Section A Mathematics · 2001
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.