Regularity and convergence of critical points of an Ambrosio-Tortorelli functional with linear growth and of its Γ-limit

In the one-dimensional setting we consider an Ambrosio-Tortorelli functional Fε(u,v) which has linear growth with respect to u′. We prove that under suitable conditions on the fidelity term, minimizers and critical points of Fε are Sobolev regular, and that the same is true for the Γ-limit F of Fε. As a corollary, we obtain that the functional Aw(u) computing the length of the generalized graph of a function of bounded variation u, under the same conditions on the fidelity term, admits a unique minimizer of class C1. This partially solves a conjecture by De Giorgi [16] in the one-dimensional case.