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| Main Authors: | , |
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| Format: | Preprint |
| Published: |
2022
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2202.09295 |
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| _version_ | 1866915333967708160 |
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| author | Minne, Andreas Tewodrose, David |
| author_facet | Minne, Andreas Tewodrose, David |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2202_09295 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | Symmetrized and non-symmetrized Asymptotic Mean Value Laplacian in metric measure spaces Minne, Andreas Tewodrose, David Analysis of PDEs 35J05, 47F10, 31C12 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. |
| title | Symmetrized and non-symmetrized Asymptotic Mean Value Laplacian in metric measure spaces |
| topic | Analysis of PDEs 35J05, 47F10, 31C12 |
| url | https://arxiv.org/abs/2202.09295 |