分段刚性函数和$GSBD$上泛函的下半连续性
Lower semicontinuity for functionals defined on piecewise rigid functions and on $GSBD$
摘要：
在这项工作中，我们提供了一个表征结果的下半连续性的表面能定义在分段刚性函数，即，函数是分段仿射的Caccioppoli分区，其中每个组件的衍生物是一个反对称矩阵。该特征通过称为$BD$ -椭圆率的积分条件来实现，该积分条件符合Ambrosio和Braides定义的$BV$ -椭圆率的精神。通过具体的例子，我们表明，这种新的概念实际上是更强的相比，其$BV$类似物。我们进一步给出了一个隐含$BD$ -椭圆性的充分条件，我们称之为对称联合凸性。可以针对与应用相关的某些类别的表面能明确地检查该概念，例如，变分断裂模型。最后，我们给出了一个直接的证明，证明了在更大的$GSBD^p$函数空间上，具有对称联合凸被积函数的表面能也是下连续的.
英文摘要：
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.
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