On the convergence of critical points of the

This work is devoted to study 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 & lowast;,1) as epsilon -> 0, where u & lowast; is a special function of bounded variation. Assuming further the Ambrosio-Tortorelli energy of (u epsilon,v epsilon) to converge to the Mumford-Shah energy of u & lowast;, the later is shown to be a critical point with respect to inner variations of the Mumford-Shah functional. As a byproduct, the second inner variation is also shown to pass to the limit. To establish these convergence results, interior (C infinity) 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.