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

Andreas Minne, David Tewodrose

Research output: Contribution to journalArticlepeer-review

Abstract

The asymptotic mean value Laplacian - AMV Laplacian - extends the Laplace operator from 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 - where the two operators typically differ - and provide explicit formulae for these operators, including points where the weight vanishes.

Original languageEnglish
JournalProceedings of the Royal Society of Edinburgh Section A: Mathematics
DOIs
Publication statusAccepted/In press - 2023

Bibliographical note

Funding Information:
The first author is grateful for the support by the Knut and Alice Wallenberg Foundation, Project KAW 2015.0380. The second author is supported by Laboratoire de Mathématiques Jean Leray via the project Centre Henri Lebesgue and Fédération de recherche Mathématiques de Pays de Loire via the project Ambition Lebesgue Loire. The authors thank Giorgio Stefani and Luca Rizzi for pointing out reference []. They are also both grateful for precious remarks made by the anonymous referees. The second author thanks Gilles Carron for inspiring discussions and Luca Rizzi for his invitation at SISSA Trieste-funded by the European Research Council (ERC) under the European Union's Horizon 2020 research and innovation programme (grant agreement No. 945655)—where helpful discussions took place.

Publisher Copyright:
Copyright © The Author(s), 2023. Published by Cambridge University Press on behalf of The Royal Society of Edinburgh.

Keywords

  • average integral
  • Laplace operator
  • mean value property
  • metric measure space

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