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

TL;DR

This paper investigates the asymptotic behavior of critical points of the Ambrosio-Tortorelli functional, showing their convergence to critical points of the Mumford-Shah functional under certain energy conditions, using varifold theory.

Contribution

It demonstrates that critical points of the Ambrosio-Tortorelli functional converge to critical points of the Mumford-Shah functional, extending understanding of their relationship in phase transition models.

Findings

Critical points converge in $L^2$ to a Mumford-Shah critical point.

The second inner variation passes to the limit.

Interior and boundary regularity results are established.

Abstract

This work is devoted to study the asymptotic behavior of critical points $\{(u_\varepsilon,v_\varepsilon)\}_{\varepsilon>0}$ 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.