Regular Approximation of Free-Discontinuity Problems

We consider a class of smooth local nonconvex functionals defined on W 2,2 (Ω), depending on a small parameter ε and we prove that they converge, as ε tends to 0, to a functional F(u,Ω) with a bulk density depending on the gradient of u and a surface energy concentrated on the jump set of u. This provides a new alternative to the approximation of free discontinuity problems, which applies in particular to the Mumford–Shah model.