Symmetrized and non-symmetrized Asymptotic Mean Value Laplacian in metric measure spaces

TL;DR

This paper investigates the symmetric and non-symmetric asymptotic mean value Laplacians in metric measure spaces, identifying conditions under which they coincide and providing explicit formulas in weighted Euclidean domains.

Contribution

It introduces a symmetric version of the AMV Laplacian, analyzes when it matches the non-symmetric version, and derives explicit formulas in weighted Euclidean spaces.

Findings

Symmetric and non-symmetric AMV Laplacians coincide on Riemannian and certain sub-Riemannian manifolds.

They are also identical on a broad class of metric measure spaces including locally Ahlfors regular spaces.

Explicit formulas are provided for weighted Euclidean domains, even at points where weights vanish.

Abstract

The asymptotic mean value Laplacian - AMV Laplacian - extends the Laplace operator from $\mathbb{R}^n$ to metric measure spaces through limits of averaging integrals. The AMV Laplacian is however not a symmetric operator in general. In this paper therefore a symmetric version of the AMV Laplacian is considered, and focus lies on when the symmetric and non-symmetric AMV operators coincide. Besides Riemannian and 3D contact sub-Riemannian manifolds, we show that they are identical on a large class of metric measure spaces including locally Ahlfors regular spaces with vanishing metric-measure boundary. In addition, we study the context of weighted domains of $\mathbb{R}^n$ where the two operators typically differ, and provide concrete formulae for these operators also at points where the weight vanishes.