有界变形的哪些特殊函数有界变差？
Which special functions of bounded deformation have bounded variation?
摘要：
有界变形函数（$BD$）是在几何线性背景下研究断裂和损伤问题时自然产生的。它们与有界变差函数（$BV$）有关，但不太清楚。我们在这里讨论的关系，$BV$额外的正则性假设下，这可能需要定期的应变部分有较高的可积性或跳跃集有有限的面积或康托部分消失。在积极的方面，我们证明了在Caccioppoli划分上分段仿射的$BD$函数在$GSBV$中，并且我们证明了$SBD^p$函数是近似连续的${\mathcal{H}}^{n-1}$ -a.e.远离跳伞场地在消极的一面，我们构造了一个函数，这是$BD$，但不是在$BV$，并有分布应变只包括一个跳跃部分，和一个只有一个康托部分组成的分布应变。
英文摘要：
Functions of bounded deformation ($BD$) arise naturally in the study of fracture and damage in a geometrically linear context. They are related to functions of bounded variation ($BV$), but are less well understood. We discuss here the relation to $BV$ under additional regularity assumptions, which may require the regular part of the strain to have higher integrability or the jump set to have finite area or the Cantor part to vanish. On the positive side, we prove that $BD$ functions which are piecewise affine on a Caccioppoli partition are in $GSBV$, and we prove that $SBD^p$ functions are approximately continuous ${\mathcal{H}}^{n-1}$-a.e. away from the jump set. On the negative side, we construct a function which is $BD$ but not in $BV$ and has distributional strain consisting only of a jump part, and one which has a distributional strain consisting of only a Cantor part.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
