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| Main Author: | |
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| Format: | Preprint |
| Published: |
2026
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2601.16063 |
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| _version_ | 1866915747781935104 |
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| author | Shi, Kehan |
| author_facet | Shi, Kehan |
| contents | This paper studies a class of $p$-Laplacian equations on point clouds that arise from hypergraph learning in a semi-supervised setting. Under the assumption that the point clouds consist of independent random samples drawn from a bounded domain $Ω\subset\mathbb{R}^d$, we investigate the asymptotic behavior of the solutions as the number of data points tends to infinity, with the number of labeled points remains fixed. We show, for any $p>d$ in the viscosity solution framework, that the continuum limit is a weighted $p$-Laplacian equation subject to mixed Dirichlet and Neumann boundary conditions. The result provides a new discretization of the $p$-Laplacian on point clouds. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2601_16063 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Continuum limit of hypergraph $p$-Laplacian equations on point clouds Shi, Kehan Analysis of PDEs This paper studies a class of $p$-Laplacian equations on point clouds that arise from hypergraph learning in a semi-supervised setting. Under the assumption that the point clouds consist of independent random samples drawn from a bounded domain $Ω\subset\mathbb{R}^d$, we investigate the asymptotic behavior of the solutions as the number of data points tends to infinity, with the number of labeled points remains fixed. We show, for any $p>d$ in the viscosity solution framework, that the continuum limit is a weighted $p$-Laplacian equation subject to mixed Dirichlet and Neumann boundary conditions. The result provides a new discretization of the $p$-Laplacian on point clouds. |
| title | Continuum limit of hypergraph $p$-Laplacian equations on point clouds |
| topic | Analysis of PDEs |
| url | https://arxiv.org/abs/2601.16063 |