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

We consider a family $\{(u_{\epsilon },v_{\epsilon }){\}}_{\epsilon >0}$ of critical points of the Ambrosio-Tortorelli functional. Assuming a uniform energy bound, the sequence $\{(u_{\epsilon },v_{\epsilon }){\}}_{\epsilon >0}$ converges in $L^2(\mathrm{\Omega })$ to a limit $(u,1)$ as $\epsilon \to 0$, where $u$ is in $SB{V}^2(\mathrm{\Omega })$. It was previously shown that if the full Ambrosio-Tortorelli energy associated to $(u_{\epsilon },v_{\epsilon })$ 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.