Caloric functions and boundary regularity for the fractional Laplacian in Lipschitz open sets
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arXiv
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| Autores principales: | , , |
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| Formato: | Preprint |
| Publicado: |
2024
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| _version_ | 1866914878773526528 |
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| author | Armstrong, Gavin Bogdan, Krzysztof Rutkowski, Artur |
| author_facet | Armstrong, Gavin Bogdan, Krzysztof Rutkowski, Artur |
| contents | We give Martin representation of nonnegative functions caloric with respect to the fractional Laplacian in Lipschitz open sets. The caloric functions are defined in terms of the mean value property for the space-time isotropic $α$-stable Lévy process. To derive the representation, we first establish the existence of the parabolic Martin kernel. This involves proving new boundary regularity results for both the fractional heat equation and the fractional Poisson equation with Dirichlet exterior conditions. Specifically, we demonstrate that the ratio of the solution and the Green function is Hölder continuous up to the boundary. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2403_03840 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Caloric functions and boundary regularity for the fractional Laplacian in Lipschitz open sets Armstrong, Gavin Bogdan, Krzysztof Rutkowski, Artur Analysis of PDEs Functional Analysis Probability 35S16, 60J50, 35C15 We give Martin representation of nonnegative functions caloric with respect to the fractional Laplacian in Lipschitz open sets. The caloric functions are defined in terms of the mean value property for the space-time isotropic $α$-stable Lévy process. To derive the representation, we first establish the existence of the parabolic Martin kernel. This involves proving new boundary regularity results for both the fractional heat equation and the fractional Poisson equation with Dirichlet exterior conditions. Specifically, we demonstrate that the ratio of the solution and the Green function is Hölder continuous up to the boundary. |
| title | Caloric functions and boundary regularity for the fractional Laplacian in Lipschitz open sets |
| topic | Analysis of PDEs Functional Analysis Probability 35S16, 60J50, 35C15 |
| url | https://arxiv.org/abs/2403.03840 |