A note on the one-dimensional critical points of the Ambrosio-Tortorelli functional

International audience ; This note addresses the question of convergence of critical points of the Ambrosio-Tortorelli functional in the one-dimensional case under pure Dirichlet boundary conditions. An asymptotic analysis argument shows the convergence to two possible limits points: either a globally affine function or a step function with a single jump at the middle point of the space interval, which are both critical points of the one-dimensional Mumford-Shah functional under a Dirichlet boundary condition. As a byproduct, non minimizing critical points of the Ambrosio-Tortorelli functional satisfying the energy convergence assumption as in \cite{BMR} are proved to exist.