2022/02/18 by Andreas Minne, Minne, Andreas, David Tewodrose +1 · 1 citation
Mathematics · #31C12 #35J05 #47F10 #Analysis of PDEs (math.AP) #FOS: Mathematics #Geometric Analysis and Curvature Flows #Nonlinear Partial Differential Equations #Numerical methods in inverse problems
paper · pdf · doi:10.48550/arxiv.2202.09295
openalex publication_date 2022/02/18 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
The asymptotic mean value Laplacian - AMV Laplacian - extends the Laplace operator from ℝn 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 ℝn where the two operators typically differ, and provide concrete formulae for these operators also at points where the weight vanishes.