Spectral properties of symmetrized AMV operators
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arXiv
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| Hauptverfasser: | , |
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| Format: | Preprint |
| Veröffentlicht: |
2024
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| _version_ | 1866911094062186496 |
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| author | Dias, Manuel Tewodrose, David |
| author_facet | Dias, Manuel Tewodrose, David |
| contents | The symmetrized Asymptotic Mean Value Laplacian $\tildeΔ$, obtained as limit of approximating operators $\tildeΔ_r$, is an extension of the classical Euclidean Laplace operator to the realm of metric measure spaces. We show that, as $r \downarrow 0$, the operators $\tildeΔ_r$ eventually admit isolated eigenvalues defined via min-max procedure on any compact locally Ahlfors regular metric measure space. Then we prove $L^2$ and spectral convergence of $\tildeΔ_r$ to the Laplace--Beltrami operator of a compact Riemannian manifold, imposing Neumann conditions when the manifold has a non-empty boundary. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2411_10202 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Spectral properties of symmetrized AMV operators Dias, Manuel Tewodrose, David Analysis of PDEs Metric Geometry Spectral Theory 58J50, 35J05, 30L99, 35P05 The symmetrized Asymptotic Mean Value Laplacian $\tildeΔ$, obtained as limit of approximating operators $\tildeΔ_r$, is an extension of the classical Euclidean Laplace operator to the realm of metric measure spaces. We show that, as $r \downarrow 0$, the operators $\tildeΔ_r$ eventually admit isolated eigenvalues defined via min-max procedure on any compact locally Ahlfors regular metric measure space. Then we prove $L^2$ and spectral convergence of $\tildeΔ_r$ to the Laplace--Beltrami operator of a compact Riemannian manifold, imposing Neumann conditions when the manifold has a non-empty boundary. |
| title | Spectral properties of symmetrized AMV operators |
| topic | Analysis of PDEs Metric Geometry Spectral Theory 58J50, 35J05, 30L99, 35P05 |
| url | https://arxiv.org/abs/2411.10202 |