# Asymptotic Mean Value Laplacian in Metric Measure Spaces

created by tewodrose on 09 Dec 2019
modified on 19 Nov 2020

[BibTeX]

Published Paper

Inserted: 9 dec 2019
Last Updated: 19 nov 2020

Journal: Journal of Mathematical Analysis and Applications
Volume: 491
Year: 2020

ArXiv: 1912.00259 PDF

Abstract:

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.

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