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

Abstract

This work is devoted to studying the asymptotic behavior of critical points $\{(u_{\epsilon },v_{\epsilon }){\}}_{\epsilon >0}$ of the Ambrosio–Tortorelli functional. Under a uniform energy bound assumption, the usual $\Gamma$-convergence theory ensures that $(u_{\epsilon },v_{\epsilon })$ converges in the $L^2$-sense to some $(u_{\ast },1)$ as $\epsilon \to 0$, where $u_{\ast }$ is a special function of bounded variation. Assuming further that the Ambrosio–Tortorelli energy of $(u_{\epsilon },v_{\epsilon })$ converges to the Mumford–Shah energy of $u_{\ast }$, the latter is shown to be a critical point with respect to inner variations of the Mumford–Shah functional. As a by-product, the second inner variation is also shown to pass to the limit. To establish these convergence results, interior $({\mathcal{C}}^{\infty })$ regularity and boundary regularity for Dirichlet boundary conditions are first obtained for a fixed parameter $\epsilon >0$. The asymptotic analysis is then performed by means of varifold theory in the spirit of scalar phase transition problems.