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This work is devoted to studying the asymptotic behavior of critical points {(uε, νε)}ε>0 of the Ambrosio-Tortorelli functional. Under a uniform energy bound assumption, the usual Γ- convergence theory ensures that (uε, νε) converges in the L²-sense to some (u*, 1) as ɛ → 0, where u* is a special function of bounded variation. Assuming further that the Ambrosio-Tortorelli energy of (uε, vε) converges to the Mumford-Shah energy of u*, 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 (∞) regularity and boundary regularity for Dirichlet boundary conditions are first obtained for a fixed parameter & > 0. The asymptotic analysis is then performed by means of varifold theory in the spirit of scalar phase transition problems. [ABSTRACT FROM AUTHOR]

In this paper we prove a C1,α regularity result for minimizers of the planar Griffith functional arising from a variational model of brittle fracture. We prove that any isolated connected component of the crack, the singular set of a minimizer, is locally a C1,α curve outside a set of zero Hausdorff measure. [ABSTRACT FROM AUTHOR]