Rectifiability of entropy defect measures in a micromagnetics model

The rectifiability of the entropy defect measure for Burgers equation

We consider bounded weak solutions to the Burgers equation for which every entropy dissipation is representable by a measure and we prove that all these measures are concentrated on the graphs of countably many Lipschitz curves. The main tool is the Lagrangian representation, which is an extension of the method of characteristics to the non-smooth setting.

取向度可变的液晶中线缺陷的可矫正性

Rectifiability of line defects in liquid crystals with variable degree of orientation

在[AHL] Hardt，Lin和作者证明了双相液晶修正Ericksen能量极小的缺陷集局部由孤立点和具有多个交叉点的Hölder连续曲线的有限并集组成。在这篇文章中，我们证明了亏集中的每一条Hölder连续曲线都是有限长的。因此，缺陷集是局部可纠正的。在大多数情况下，证明如下的工作德Lellis，Marchese，Spadaro和Valtorta [DLMSV]调和$\mathcal{Q}$值地图密切。[AHL]中的爆破分析使我们能够简化[DLMSV]中的覆盖参数，并以几何方式局部估计线缺陷的长度。

Dimensional estimates and rectifiability for measures satisfying linear PDE constraints

We establish the rectifiability of measures satisfying a linear PDE constraint. The obtained rectifiability dimensions are optimal for many usual PDE operators, including all first-order systems and all second-order scalar operators. In particular, our general theorem provides a new proof of the rectifiability results for functions of bounded variations (BV) and functions of bounded deformation (BD). For divergence-free tensors we obtain refinements and new proofs of several known results on the rectifiability of varifolds and defect measures.

An effective model for boundary vortices in thin-film micromagnetics

Ferromagnetic materials are governed by a variational principle which is nonlocal, nonconvex and multiscale. The main object is given by a unit-length three-dimensional vector field, the magnetization, that corresponds to the stable states of the micromagnetic energy. Our aim is to analyze a thin film regime that captures the asymptotic behavior of boundary vortices generated by the magnetization and their interaction energy. This study is based on the notion of "global Jacobian" detecting the topological defects that a priori could be located in the interior and at the boundary of the film. A…

惩罚沿两个独立方向同时振荡的非凸泛函：缺陷测度的结构

Non-convex functionals penalizing simultaneous oscillations along two independent directions: structure of the defect measure

本文研究惩罚沿两个独立方向同时振荡的非凸能量，刻画有限能量函数缺陷测度的可求积性，并证明不同参数与维数情形下的结构结果。

广义$XY$模型的$Γ$ -收敛性分析：分数涡和弦缺陷

$Γ$-convergence analysis of a generalized $XY$ model: fractional vortices and string defects

提出并分析了一个广义的二维$XY$模型，该模型的相互作用势具有$n$权势阱，描述了系统相应的对称性.当晶格间距为零时，我们通过$Γ$ -收敛得到了该模型的离散到连续极限。在我们处理的能量制度，渐近基态表现出分数涡，连接弦缺陷。该$Γ$ -限制考虑到这两个贡献，通过重整化能量，取决于分数涡的配置，和表面能，成比例的弦的长度。我们的模型以一种简单的方式描述了物理学和材料科学中出现的几个拓扑奇点。其中，液晶中的向错和弦缺陷，微磁学中的分数涡旋和畴壁，晶体塑性中的部分位错和层错。