论文标题
在分段刚性函数和$ GSBD $上定义的功能的较低的半连续性
Lower semicontinuity for functionals defined on piecewise rigid functions and on $GSBD$
论文作者
论文摘要
在这项工作中，我们为在分段刚性函数上定义的表面能的较低半接对性提供了表征，即在caccioppoli分区上分散仿射的函数，其中每个组件中的衍生物是偏斜的对称矩阵。这种表征是通过不可或缺的条件（称为$ bd $ ellipticity）来实现的，它本着Ambrosio和Braides定义的$ bv $ ellipticity的精神。通过具体示例，我们表明，与其$ bv $类似物相比，这种新颖的概念实际上更强。我们进一步提供了足够的条件，这意味着$ bd $ ellipticity我们称之为对称的关节凸度。可以明确检查该概念的某些类别的表面能量，这些表面能与应用相关，例如差异骨折模型。最后，我们直接证明，具有对称共同凸内集成的表面能在$ GSBD^p $函数的较大空间上也较低。
In this work, we provide a characterization result for lower semicontinuity of surface energies defined on piecewise rigid functions, i.e., functions which are piecewise affine on a Caccioppoli partition where the derivative in each component is a skew symmetric matrix. This characterization is achieved by means of an integral condition, called $BD$-ellipticity, which is in the spirit of $BV$-ellipticity defined by Ambrosio and Braides. By specific examples we show that this novel concept is in fact stronger compared to its $BV$ analog. We further provide a sufficient condition implying $BD$-ellipticity which we call symmetric joint convexity. This notion can be checked explicitly for certain classes of surface energies which are relevant for applications, e.g., for variational fracture models. Finally, we give a direct proof that surface energies with symmetric jointly convex integrands are lower semicontinuous also on the larger space of $GSBD^p$ functions.
