We use the mean value property in an asymptotic way to provide a notion of a pointwise Laplacian, called AMV Laplacian, that we study in several contexts including the Heisenberg group and weighted Lebesgue measures. We focus especially on a class of metric measure spaces including intersecting submanifolds of $\mathbb{R}^n$, a context in which our notion brings new insights; the Kirchhoff law appears as a special case. In the general case, we also prove a maximum and comparison principle, as well as a Green-type identity for a related operator.

中文翻译：

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度量空间中的渐近平均值拉普拉斯算子

我们以渐近的方式使用平均值属性来提供点式拉普拉斯算子的概念，称为 AMV 拉普拉斯算子，我们在包括海森堡群和加权 Lebesgue 测度在内的多种情况下研究该概念。我们特别关注一类度量度量空间，包括 $\mathbb{R}^n$ 的相交子流形，我们的概念在这个上下文中带来了新的见解；基尔霍夫定律是一个特例。在一般情况下，我们还证明了最大值和比较原理，以及相关运算符的格林类型恒等式。