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Symmetrized and non-symmetrizedasymptotic mean value Laplacian in metric measure spaces

Part of: Partial differential equations Analysis on metric spaces Elliptic equations and systems

Published online by Cambridge University Press:  28 November 2023

Andreas Minne Open the ORCID record for Andreas Minne [Opens in a new window]  and
David Tewodrose Open the ORCID record for David Tewodrose [Opens in a new window]
Andreas Minne
Affiliation:
KTH Royal Institute of Technology, 100 44 Stockholm, Sweden (minne@kth.se)
David Tewodrose
Affiliation:
Laboratoire de Mathématiques Jean Leray, Nantes Université, UMR CNRS 6629, 2 rue de la Houssiniére BP 92208, F-44322 Nantes Cedex 3, France (david.tewodrose@vub.be)
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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. 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 $\mathbb {R}^n$—where the two operators typically differ—and provide explicit formulae for these operators, including points where the weight vanishes.

Keywords

Laplace operatormetric measure spacemean value propertyaverage integral

MSC classification

Primary: 35J05: Laplacian operator, reduced wave equation (Helmholtz equation), Poisson equation
Secondary: 35B05: Oscillation, zeros of solutions, mean value theorems, etc. 30L99: None of the above, but in this section
Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , First View , pp. 1 - 38
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Copyright
Copyright © The Author(s), 2023. Published by Cambridge University Press on behalf of The Royal Society of Edinburgh

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