Asymptotic Mean Value Laplacian in Metric Measure Spaces

TL;DR

This paper introduces the AMV Laplacian, a new pointwise Laplacian concept based on asymptotic mean value properties, applicable in various metric measure spaces including submanifolds and the Heisenberg group.

Contribution

It defines and studies the AMV Laplacian in metric measure spaces, establishing fundamental properties and extending classical Laplacian concepts to new contexts.

Findings

Introduces the AMV Laplacian using asymptotic mean value properties.

Proves maximum and comparison principles for the AMV Laplacian.

Derives a Green-type identity in the general setting.

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.