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 [5]. 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 $GSB{D}^p$ functions.

在这项工作中，我们为分段刚性函数定义的表面能的较低半连续性提供了一个表征结果，分段刚性函数即在Caccioppoli分区上分段仿射的函数，其中每个分量中的导数都是倾斜对称矩阵。这种表征是通过一积分条件，称为来实现BD-椭圆率，这是在精神BV -ellipticity通过安布罗西奥和Braides [5]定义的。通过具体的例子，我们表明，与BV类似物相比，这种新颖的概念实际上更强大。我们还提供了一个充分的条件来暗示BD椭圆率，我们称其为对称联合凸度。可以针对某些与应用相关的表面能类别（例如，变化裂缝模型）明确检查该概念。最后，我们给出一个直接的证明，即对称对称的凸凸被积体的表面能在较大的空间上也是较低的半连续的$G\mathrm{小号}乙d^p$ 职能。