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| Autores principales: | , , , , |
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| Formato: | Preprint |
| Publicado: |
2025
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| Materias: | |
| Acceso en línea: | https://arxiv.org/abs/2504.09976 |
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| _version_ | 1866918382364786688 |
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| author | Arcoya, David Dipierro, Serena Lippi, Edoardo Proietti Sportelli, Caterina Valdinoci, Enrico |
| author_facet | Arcoya, David Dipierro, Serena Lippi, Edoardo Proietti Sportelli, Caterina Valdinoci, Enrico |
| contents | We present an existence and uniqueness result for weak solutions of Dirichlet boundary value problems governed by a nonlocal operator in divergence form and in the presence of a datum which is assumed to belong only to $L^1(Ω)$ and to be suitably dominated.
We also prove that the solution that we find converges, as $s\nearrow 1$, to a solution of the local counterpart problem, recovering the classical result as a limit case. This requires some nontrivial customized uniform estimates and representation formulas, given that the datum is only in $L^1(Ω)$ and therefore the usual regularity theory cannot be leveraged to our benefit in this framework.
The limit process uses a nonlocal operator, obtained as an affine transformation of a homogeneous kernel, which recovers, in the limit as $s\nearrow 1$, every classical operator in divergence form. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2504_09976 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Nonlocal operators in divergence form and existence theory for integrable data Arcoya, David Dipierro, Serena Lippi, Edoardo Proietti Sportelli, Caterina Valdinoci, Enrico Analysis of PDEs We present an existence and uniqueness result for weak solutions of Dirichlet boundary value problems governed by a nonlocal operator in divergence form and in the presence of a datum which is assumed to belong only to $L^1(Ω)$ and to be suitably dominated. We also prove that the solution that we find converges, as $s\nearrow 1$, to a solution of the local counterpart problem, recovering the classical result as a limit case. This requires some nontrivial customized uniform estimates and representation formulas, given that the datum is only in $L^1(Ω)$ and therefore the usual regularity theory cannot be leveraged to our benefit in this framework. The limit process uses a nonlocal operator, obtained as an affine transformation of a homogeneous kernel, which recovers, in the limit as $s\nearrow 1$, every classical operator in divergence form. |
| title | Nonlocal operators in divergence form and existence theory for integrable data |
| topic | Analysis of PDEs |
| url | https://arxiv.org/abs/2504.09976 |