论文标题
关于Ambrosio tortorelli功能的临界点的收敛性
On the Convergence of critical points of the Ambrosio-Tortorelli functional
论文作者
论文摘要
这项工作致力于研究临界点的渐近行为$ \ {（（u_ \ varepsilon，v_ \ varepsilon）\} _ {\ varepsilon> 0} $ ambrosio-Tortorelli函数的$。 Under a uniform energy bound assumption, the usual $Γ$-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.假设$（u_ \ varepsilon，v_ \ varepsilon）$进一步的ambrosio-tortorelli能量收敛到$ u _*$的mumford-shah能量，则后来证明是与mumford-shah功能的内部变化相关的关键点。作为副产品，第二个内部变化也显示为极限。为了建立这些收敛结果，首先获得了固定参数$ \ varepsilon> 0 $的内部（$ \ mathscr {c}^\ infty $）的规律性和dirichlet边界条件的边界规律性。然后，在标量相变问题的精神上通过Varifold理论进行渐近分析。
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 $Γ$-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.
