抛物型安布罗西奥 - 托尔托雷利系统解的存在性、唯一性和正则性

关于 $\mathbb S^1$ 值映射的跳跃最小化提升及其与 Ambrosio-Tortorelli 型 $Γ$-极限的联系

On jump minimizing liftings for $\mathbb S^1$-valued maps and connections with Ambrosio-Tortorelli-type $Γ$-limits

该研究刻画 $\mathbb S^1$ 值映射的 Ambrosio-Tortorelli 型泛函在不同定义域下的两类 $Γ$-极限，并通过提升序列的紧性结果建立非局部极限与跳跃最小化提升存在性的联系。

Second-order edge-penalization in the Ambrosio-Tortorelli functional

We propose and study two variants of the Ambrosio-Tortorelli functional where the first-order penalization of the edge variable $v$ is replaced by a second-order term depending on the Hessian or on the Laplacian of $v$, respectively. We show that both the variants as above provide an elliptic approximation of the Mumford-Shah functional in the sense of $Γ$-convergence. In particular the variant with the Laplacian penalization can be implemented without any difficulties compared to the standard Ambrosio-Tortorelli functional. The computational results indicate several advantages however. First …

Ambrosio-Tortorelli 泛函梯度流的存在性、唯一性及 $L^2_t (H_x ^2) \cap L^\infty_t (H^1_x) \cap H^1_t (L^2_x)$ 正则性

Existence, uniqueness and $L^2_t (H_x ^2) \cap L^\infty_t (H^1_x) \cap H^1_t (L^2_x) $ regularity of the gradient flow of the Ambrosio-Tortorelli functional

本文研究二维固定参数下 Ambrosio-Tortorelli 泛函的梯度流，借助新的关键 $L^2_t(H_x^2)$ 估计技术，证明其全局存在性、唯一性及所需混合时空正则性。

Random finite-difference discretizations of the Ambrosio-Tortorelli functional with optimal mesh size

We propose and analyze a finite-difference discretization of the Ambrosio-Tortorelli functional. It is known that if the discretization is made with respect to an underlying periodic lattice of spacing $δ$, the discretized functionals $Γ$-converge to the Mumford-Shah functional only if $δ\ll\varepsilon$, $\varepsilon$ being the elliptic approximation parameter of the Ambrosio-Tortorelli functional. Discretizing with respect to stationary, ergodic and isotropic random lattices we prove this $Γ$-convergence result also for $δ\sim\varepsilon$, a regime at which the discretization with respect to …

$\Gamma$-收敛：图像分割的Ambrosio-Tortorelli逼近格式

$Γ$-Convergence of an Ambrosio-Tortorelli approximation scheme for image segmentation

本文证明了Ambrosio-Tortorelli逼近在分片光滑设置下对Mumford-Shah泛函的$\Gamma$-收敛，从而为图像分割中该逼近的极小元收敛性提供了理论保证。

Phase-field approximation of functionals defined on piecewise-rigid maps

We provide a variational approximation of Ambrosio-Tortorelli type for brittle fracture energies of piecewise-rigid solids. Our result covers both the case of geometrically nonlinear elasticity and that of linearised elasticity.

Approximation of the Mumford-Shah Functional by Phase Fields of Bounded Variation

In this paper we introduce a new phase field approximation of the Mumford-Shah functional similar to the well-known one from Ambrosio and Tortorelli. However, in our setting the phase field is allowed to be a function of bounded variation, instead of an $H^1$-function. In the context of image segmentation, we also show how this new approximation can be used for numerical computations, which contains a total variation minimization of the phase field variable, as it appears in many problems of image processing. A comparison to the classical Ambrosio-Tortorelli approximation, where the phase fiel…

$Γ$-收敛自由不连续问题中的圆值映射线性情形

$Γ$-convergence of free discontinuity problems for circle-valued maps in the linear regime

本文研究了圈值映射在线性增长情形下的$Γ$-收敛性，揭示了非局部极限的形成机制，并扩展了之前二次情形的分析。

Phase-field approximation for a class of cohesive fracture energies with an activation threshold

We study the $Γ$-limit of Ambrosio-Tortorelli-type functionals $D_\varepsilon(u,v)$, whose dependence on the symmetrised gradient $e(u)$ is different in $\mathbb{A} u$ and in $e(u)-\mathbb{A} u$, for a $\mathbb{C}$-elliptic symmetric operator $\mathbb{A}$, in terms of the prefactor depending on the phase-field variable $v$. This is intermediate between an approximation for the Griffith brittle fracture energy and the one for a cohesive energy by Focardi and Iurlano. In particular we prove that $G(S)BD$ functions with bounded $\mathbb{A}$-variation are $(S)BD$.

含裂纹相关耗散的动态断裂相场模型解的存在性

Existence of solutions to a phase-field model of dynamic fracture with a crack-dependent dissipation

该研究提出含裂纹尖端速度耗散效应的Ambrosio–Tortorelli型动态断裂相场模型，通过调整已有时间离散格式证明满足Griffith能量耗散平衡的裂纹演化解存在，并分析了无耗散的相关动态相场模型。

变分断裂模型梯度流结构的物理起源是什么？

What is the physical origin of the gradient flow structure of variational fracture models?

本文通过比较 Griffith 型断裂模型与不可逆相场模型的耗散结构并分析行波解，揭示相场模型小时间松弛参数对应断裂能速度依赖性速率的物理起源。

Existence and Uniqueness of the Motion by Curvature of regular networks

We prove existence and uniqueness of the motion by curvatureof networks in $\mathbb{R}^n$ when the initial datum is of class $W^{2-\frac{2}{p}}_p$, with triple junction where the unit tangent vectors to the concurring curves form angles of $120$ degrees. Moreover we investigated the regularization effect due to the parabolic nature of the system. An application of this wellposedness result is a new proof of Theorem 3.18 in "Motion by Curvature of Planar Networks" by Mantegazza-Novaga-Tortorelli where the possible behaviors of the solutions at the maximal time of existence are described. Our st…

q-heat flow and the gradient flow of the Renyi entropy in the p-Wasserstein space

Based on the idea of a recent paper by Ambrosio-Gigli-Savaré in Invent. Math. (2013), we show that flow of the $q$-Cheeger energy, called $q$-heat flow, solves the gradient flow problem of the Renyi entropy functional in the $p$-Wasserstein. For that, a further study of the $q$-heat flow is presented including a condition for its mass preservation. Under a convexity assumption on the upper gradient, which holds for all $q\ge2$, one gets uniqueness of the gradient flow and the two flows can be identified. Smooth solution of the $q$-heat flow are solution the parabolic q-Laplace equation, i.e. $…