We provide relaxation for not lower semicontinuous supremal functionals defined on vectorial Lipschitz functions, where the Borel level convex density depends only on the gradient. The connection with indicator functionals is also enlightened, thus extending previous lower semicontinuity results in that framework. Finally, we discuss the power law approximation of supremal functionals, with nonnegative, coercive densities having explicit dependence also on the spatial variable, and satisfying minimal measurability assumptions.

中文翻译：

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超泛函向量设置和幂律近似中的松弛结果

我们为矢量 Lipschitz 函数上定义的非下半连续超函数提供松弛，其中 Borel 级凸密度仅取决于梯度。与指标泛函的联系也得到启发，从而扩展了该框架中先前的较低半连续性结果。最后，我们讨论了超函数的幂律近似，其中非负、强制密度也显式依赖于空间变量，并满足最小可测量性假设。