Calculus of Variations and Geometric Measure Theory

A. Minne - D. Tewodrose

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

created by tewodrose on 24 Feb 2022
modified on 25 Oct 2023


Accepted Paper

Inserted: 24 feb 2022
Last Updated: 25 oct 2023

Journal: Proceedings of the Royal Society of Edinburgh Section A: Mathematics
Year: 2022

ArXiv: 2202.09295 PDF


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.