Calculus of Variations and Geometric Measure Theory

J. F. Babadjian - V. Millot - R. Rodiac

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

created by babadjian on 07 Oct 2022
modified on 28 Nov 2022


Submitted Paper

Inserted: 7 oct 2022
Last Updated: 28 nov 2022

Year: 2022

ArXiv: 2210.03533 PDF


This work is devoted to study the asymptotic behavior of critical points $(u_\varepsilon,v_\varepsilon)\}$ of the Ambrosio-Tortorelli functional. Under a uniform energy bound assumption, the usual $\Gamma$-convergence theory ensures that $(u_\varepsilon,v_\varepsilon)\}$ converges in the $L^2$-sense to some $(u_*,1)$ as $\varepsilon \to 0$, where $u_*$ is a special function of bounded variation. Assuming further the Ambrosio-Tortorelli energy of $(u_\varepsilon,v_\varepsilon)\}$ to converge to the Mumford-Shah energy of $u_*$ , 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 $\mathscr{C}^\infty$ regularity and boundary regularity for Dirichlet boundary conditions are first obtained for a fixed parameter $\varepsilon>0$. The asymptotic analysis is then performed by means of varifold theory in the spirit of scalar phase transition problems.