Abstract : In this work in collaboration with Vincent Millot and Rémy Rodiac, we address the question of the convergence of critical points of the Ambrosio-Tortorelli functional, in the sense of inner variations, to those of the Mumford-Shah ones. We extend earlier results by Francfort, Le and Serfaty in the 1-dimensional case to any arbitrary dimension upon the additional assumption of the convergence of the energies. As a byproduct, we also obtain the convergence of the second inner variation, which implies the convergence of stable critical points. The proof rests on elliptic PDE and geometric measure theoretic arguments. Thanks to elliptic regularity estimates, we derive the first inner variations of the Ambrosio-Tortorelli functional which have a varifold structure. Then, we characterize the limit varifold as the rectifiable varifold associated to the jump set.

Keywords : free discontinuity problems; geometric measure theory; elliptic regularity