Abstract Chen, Torres and Ziemer ([9], 2009) proved the validity of generalized Gauss–Green formulas and obtained the existence of interior and exterior normal traces for essentially bounded divergence measure fields on sets of finite perimeter using an approximation theory through sets with a smooth boundary. However, it is known that the proof of a crucial approximation lemma contained a gap. Taking inspiration from a previous work of Chen and Torres ([7], 2005) and exploiting ideas of Vol’pert ([29], 1985) for essentially bounded fields with components of bounded variation, we present here a direct proof of generalized Gauss–Green formulas for essentially bounded divergence measure fields on sets of finite perimeter which includes the existence and essential boundedness of the normal traces. Our approach appears to be simpler since it does not require any special approximation theory for the domains and it relies only on the Leibniz rule for divergence measure fields. This freedom allows one to localize the constructions and to derive more general statements in a natural way.

中文翻译：

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关于局部本质上有界的散度测度场和局部有限周长集

摘要 Chen、Torres 和 Ziemer ([9], 2009) 证明了广义 Gauss-Green 公式的有效性，并使用近似理论通过具有平滑的边界。然而，众所周知，关键逼近引理的证明包含一个缺口。从 Chen 和 Torres ([7], 2005) 之前的工作中汲取灵感，并利用 Vol'pert ([29], 1985) 的思想对具有有界变化分量的本质有界场，我们在此提出了广义高斯的直接证明– 用于有限周长集上的基本有界散度测量场的绿色公式，其中包括法线迹的存在性和基本有界性。我们的方法似乎更简单，因为它不需要域的任何特殊近似理论，并且它仅依赖于散度度量域的莱布尼茨规则。这种自由允许人们以一种自然的方式定位结构并推导出更一般的陈述。