Nonlocal-to-local $L^p$-convergence of convolution operators with singular, anisotropic kernels
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arXiv
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| Autores principales: | , , |
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| Formato: | Preprint |
| Publicado: |
2026
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| _version_ | 1866912915673579520 |
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| author | Abels, Helmut Hurm, Christoph Knopf, Patrik |
| author_facet | Abels, Helmut Hurm, Christoph Knopf, Patrik |
| contents | We study nonlocal convolution-type operators with singular, possibly anisotropic kernels. Our main objective is to establish and quantify their nonlocal-to-local convergence to a local differential operator with natural boundary conditions, as the kernels concentrate at the origin in a suitable way. Such convergence results provide a useful tool for the physical justification of mathematical models, particularly in situations where the desired local differential operator cannot be directly derived from microscopic laws. The present work substantially extends previous results by allowing kernels with stronger singularities (comparable to those of fractional Laplacians), anisotropic and non-localized kernels, and by proving strong convergence in general $L^p$ spaces together with explicit convergence rates. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2602_18183 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Nonlocal-to-local $L^p$-convergence of convolution operators with singular, anisotropic kernels Abels, Helmut Hurm, Christoph Knopf, Patrik Analysis of PDEs Mathematical Physics Functional Analysis Primary: 47G10, Secondary: 35B40, 35J25, 47B38 We study nonlocal convolution-type operators with singular, possibly anisotropic kernels. Our main objective is to establish and quantify their nonlocal-to-local convergence to a local differential operator with natural boundary conditions, as the kernels concentrate at the origin in a suitable way. Such convergence results provide a useful tool for the physical justification of mathematical models, particularly in situations where the desired local differential operator cannot be directly derived from microscopic laws. The present work substantially extends previous results by allowing kernels with stronger singularities (comparable to those of fractional Laplacians), anisotropic and non-localized kernels, and by proving strong convergence in general $L^p$ spaces together with explicit convergence rates. |
| title | Nonlocal-to-local $L^p$-convergence of convolution operators with singular, anisotropic kernels |
| topic | Analysis of PDEs Mathematical Physics Functional Analysis Primary: 47G10, Secondary: 35B40, 35J25, 47B38 |
| url | https://arxiv.org/abs/2602.18183 |