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arXiv · 2202.09295

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

Andreas Minne·David Tewodrose
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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.

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Andreas Minne, David Tewodrose. 2022-02-18. Symmetrized and non-symmetrized Asymptotic Mean Value Laplacian in metric measure spaces. https://doi.org/10.1017/prm.2023.118

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