Critical points of the two-dimensional Ambrosio-Tortorelli functional with convergence of the phase-field energy

We consider a family \ (\ (u_, v_) \>₀\) of critical points of the Ambrosio-Tortorelli functional. Assuming a uniform energy bound, the sequence \ (u_, v_) \>₀ converges in L² () to a limit (u, 1) as 0, where \ (u\) is in \ (SBV² () \). It was previously shown that if the full Ambrosio-Tortorelli energy associated to (u_, v_) converges to the Mumford-Shah energy of u, then the first inner variation converges as well. In particular, u is a critical point of the Mumford-Shah functional in the sense of inner variations. In this work, focusing on the two-dimensional setting, we extend this result under the sole convergence of the phase-field energy to the length energy term in the Mumford-Shah functional.