Critical points of the two-dimensional Ambrosio-Tortorelli functional with convergence of the phase-field energy
Jean-François Babadjian, Martin Rakovsky, Rémy Rodiac
TL;DR
This work proves that in two dimensions, critical points of the Ambrosio–Tortorelli phase-field functional AT_ε with a uniform energy bound converge (along a subsequence) to a limit (u,1) where u ∈ SBV^2(Ω) and div(∇u)=0. Crucially, if only the phase-field energy converges to the MS surface term ℋ^1(ŵJ_u) rather than the entire AT_ε energy to MS(u), then the limit u satisfies the Mumford–Shah inner-variations condition, establishing MS criticality in the interior-variation sense. The analysis relies on a fine blow-up/defect-measure study in 2D, equi-partition of energy, and a varifold framework for the phase-field, while showing that outer (boundary) variations are not guaranteed to yield the MS Neumann condition without stronger convergence. The results illuminate the behavior of phase-field approximations for fracture and edge-detection problems under weaker convergence hypotheses and clarify what can be inferred about the limiting variational structure.
Abstract
We consider a family $\{(u_\varepsilon, v_\varepsilon)\}_{\varepsilon>0}$ of critical points of the Ambrosio-Tortorelli functional. Assuming a uniform energy bound, the sequence $\{(u_\varepsilon, v_\varepsilon)\}_{\varepsilon>0}$ converges in $L^2(Ω)$ to a limit $(u, 1)$ as $\varepsilon \to 0$, where $u$ is in $SBV^2(Ω)$. It was previously shown that if the full Ambrosio-Tortorelli energy associated to $(u_\varepsilon,v_\varepsilon)$ 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.