On the convergence of critical points of the Ambrosio-Tortorelli functional

In order to describe the behavior of an elastic material undergoing fracture, we can use a variational model and the so-called Mumford-Shah energy defined on a subspace of SBV functions. One difficulty is that the critical points of this energy are difficult to approximate by numerical methods. One can then think of approximating the Mumford-Shah energy by another energy defined on a space of more regular functions (H^1-functions): the Ambrosio-Tortorelli energy. It has been known since the pioneering work of Ambrosio-Tortorelli that the minimizers of this energy converge towards minimizers of the Mumford-Shah energy. In this talk, we will show that under an assumption of convergence of the energies, critical points of the Ambrosio-Tortorelli energy also converge to critical points of the Mumford-Shah energy. This is a joint work with Jean-François Babadjian and Vincent Millot.