论文标题
在$ 1 $可旋转电流上定义的能量的均质化
Homogenization of energies defined on $1$-rectifiable currents
论文作者
论文摘要
在本文中，我们研究了集中在线上的一类能量的均质化。在尺寸中，$ 2 $（即，在Codimension $ 1 $中）的问题减少了\ cite {ab}研究的分区能量的均质化。在那里，关键工具是用$ bv $函数在离散集中的值表示分区。在我们的一般情况下，关键成分是代表具有离散多重性的闭环，或者是在曲线上支撑的无差矩阵估计的措施，或者在晶格中具有多重性的$ 1 $ currents。在$ 3 $维度的情况下，分析此类能量的主要动机是研究晶体中的线缺陷，即所谓的脱位。
In this paper we study the homogenization of a class of energies concentrated on lines. In dimension $2$ (i.e., in codimension $1$) the problem reduces to the homogenization of partition energies studied by \cite{AB}. There, the key tool is the representation of partitions in terms of $BV$ functions with values in a discrete set. In our general case the key ingredient is the representation of closed loops with discrete multiplicity either as divergence-free matrix-valued measures supported on curves or with $1$-currents with multiplicity in a lattice. In the $3$ dimensional case the main motivation for the analysis of this class of energies is the study of line defects in crystals, the so called dislocations.
