The $L^p$ Poisson-Neumann problem and its relation to the Neumann problem
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866913402852474880 |
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| author | Feneuil, Joseph Li, Linhan |
| author_facet | Feneuil, Joseph Li, Linhan |
| contents | We introduce the $L^p$ Poisson-Neumann problem for an uniformly elliptic operator $L=-\rm{div }A\nabla$ in divergence form in a bounded 1-sided Chord Arc Domain $Ω$, which considers solutions to $Lu=h-\rm{div}\vec{F}$ in $Ω$ with zero Neumann data on the boundary for $h$ and $\vec F$ in some tent spaces. We give different characterizations of solvability of the $L^p$ Poisson-Neumann problem and its weaker variants, and in particular, we show that solvability of the weak $L^p$ Poisson-Neumann probelm is equivalent to a weak reverse Hölder inequality. We show that the Poisson-Neumman problem is closely related to the $L^p$ Neumann problem, whose solvability is a long-standing open problem. We are able to improve the extrapolation of the $L^p$ Neumann problem from Kenig and Pipher by obtaining an extrapolation result on the Poisson-Neumann problem. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2406_16735 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | The $L^p$ Poisson-Neumann problem and its relation to the Neumann problem Feneuil, Joseph Li, Linhan Analysis of PDEs 35J25 We introduce the $L^p$ Poisson-Neumann problem for an uniformly elliptic operator $L=-\rm{div }A\nabla$ in divergence form in a bounded 1-sided Chord Arc Domain $Ω$, which considers solutions to $Lu=h-\rm{div}\vec{F}$ in $Ω$ with zero Neumann data on the boundary for $h$ and $\vec F$ in some tent spaces. We give different characterizations of solvability of the $L^p$ Poisson-Neumann problem and its weaker variants, and in particular, we show that solvability of the weak $L^p$ Poisson-Neumann probelm is equivalent to a weak reverse Hölder inequality. We show that the Poisson-Neumman problem is closely related to the $L^p$ Neumann problem, whose solvability is a long-standing open problem. We are able to improve the extrapolation of the $L^p$ Neumann problem from Kenig and Pipher by obtaining an extrapolation result on the Poisson-Neumann problem. |
| title | The $L^p$ Poisson-Neumann problem and its relation to the Neumann problem |
| topic | Analysis of PDEs 35J25 |
| url | https://arxiv.org/abs/2406.16735 |