Optimal boundary regularity and Green function estimates for nonlocal equations in divergence form
Fuente:
arXiv
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| Autori principali: | , |
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| Natura: | Preprint |
| Pubblicazione: |
2024
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| _version_ | 1866916953689423872 |
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| author | Kim, Minhyun Weidner, Marvin |
| author_facet | Kim, Minhyun Weidner, Marvin |
| contents | In this article we prove for the first time the $C^s$ boundary regularity for solutions to nonlocal elliptic equations with Hölder continuous coefficients in divergence form in $C^{1,α}$ domains. So far, it was only known that solutions are Hölder continuous up to the boundary, and establishing their optimal regularity has remained an open problem in the field. Our proof is based on a delicate higher order Campanato-type iteration at the boundary, which we develop in the context of nonlocal equations and which is quite different from the local theory.
As an application of our results, we establish sharp two-sided Green function estimates in $C^{1,α}$ domains for the same class of operators. Previously, this was only known under additional structural assumptions on the coefficients and in more regular domains. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2408_12987 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Optimal boundary regularity and Green function estimates for nonlocal equations in divergence form Kim, Minhyun Weidner, Marvin Analysis of PDEs 47G20, 35B65, 35J08 In this article we prove for the first time the $C^s$ boundary regularity for solutions to nonlocal elliptic equations with Hölder continuous coefficients in divergence form in $C^{1,α}$ domains. So far, it was only known that solutions are Hölder continuous up to the boundary, and establishing their optimal regularity has remained an open problem in the field. Our proof is based on a delicate higher order Campanato-type iteration at the boundary, which we develop in the context of nonlocal equations and which is quite different from the local theory. As an application of our results, we establish sharp two-sided Green function estimates in $C^{1,α}$ domains for the same class of operators. Previously, this was only known under additional structural assumptions on the coefficients and in more regular domains. |
| title | Optimal boundary regularity and Green function estimates for nonlocal equations in divergence form |
| topic | Analysis of PDEs 47G20, 35B65, 35J08 |
| url | https://arxiv.org/abs/2408.12987 |