Abstract
The asymptotic mean value LaplacianâAMV Laplacianâextends 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. Therefore, we consider a symmetric version of the AMV Laplacian, and focus lies on when the symmetric and non-symmetric AMV Laplacians 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 suitably vanishing distortion. In addition, we study the context of weighted domains of $\mathbb {R}^n$ âwhere the two operators typically differâand provide explicit formulae for these operators, including points where the weight vanishes.
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More From: Proceedings of the Royal Society of Edinburgh: Section A Mathematics
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