AI 中文总结
该研究针对具线性增长的多凸泛函，利用Reshetnyak的连续性结果，将Acerbi-Dal Maso的连续性性质推广至BV框架，推导了严格BV松弛能量的显式公式，并获得涡旋映射L¹收敛松弛的部分结果。
AI 中文摘要
我们考虑定义在向量值函数上的多凸泛函，其模型情形为图面积泛函，并分析关于BV中严格收敛的松弛能量。在若干已知松弛面积的情形中，利用Reshetnyak的连续性结果，我们能够针对梯度 minors 中具线性增长的广泛类被积函数找到显式公式。我们初步将Acerbi-Dal Maso在Sobolev情形中观察到的连续性性质推广到BV框架。最后，得到关于涡旋映射相对于L¹收敛的松弛的部分结果。
英文摘要
We consider polyconvex functionals defined on vector valued functions, the model case being the graph area functional, and we analyze the relaxed energy with respect to the strict convergence in $BV$. In several cases where the relaxed area is known, by exploiting a continuity result by Reshetnyak we are able to find an explicit formula for wide classes of integrands with linear growth in the minors of the gradient. We preliminarily extend to the $BV$ setting a continuity property observed by Acerbi--Dal Maso in the Sobolev case. Finally, partial results concerning the relaxation of the vortex map with respect to the $L^1$ convergence are obtained.
