论文标题
非自主表面积分的$ gsbd $中的较低的半连续性
Lower semicontinuity in $GSBD$ for nonautonomous surface integrals
论文作者
论文摘要
We provide a sufficient condition for lower semicontinuity of nonautonomous noncoercive surface energies defined on the space of $GSBD^p$ functions, whose dependence on the $x$-variable is $W^{1,1}$ or even $BV$: the notion of nonautonomous symmetric joint convexity, which extends the analogous definition devised for autonomous integrands in ARXIV：2002.08133假定近似矢量场的保守性。这种情况允许在Arxiv：1512.02839中获得的$ SBV $中扩展到我们设置的非自主链公式，这是证明较低半持续结果的关键工具。可以明确检查这种新的关节凸度，以了解由不均匀材料中裂缝变异模型引起的某些类别的表面能。
We provide a sufficient condition for lower semicontinuity of nonautonomous noncoercive surface energies defined on the space of $GSBD^p$ functions, whose dependence on the $x$-variable is $W^{1,1}$ or even $BV$: the notion of nonautonomous symmetric joint convexity, which extends the analogous definition devised for autonomous integrands in arXiv:2002.08133 where the conservativeness of the approximating vector fields is assumed. This condition allows to extend to our setting a nonautonomous chain formula in $SBV$ obtained in arXiv:1512.02839, and this is a key tool in the proof of the lower semicontinuity result. This new joint convexity can be checked explicitly for some classes of surface energies arising from variational models of fractures in inhomogeneous materials.
