Localization and interpolation of parabolic $L^p$ Neumann problems
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arXiv
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| Format: | Preprint |
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2026
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| _version_ | 1866914399250284544 |
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| author | Dindoš, Martin Li, Linhan Pipher, Jill |
| author_facet | Dindoš, Martin Li, Linhan Pipher, Jill |
| contents | We show a localization estimate for local solutions to the parabolic equation $-\partial_t u+\mbox{div} (A\nabla u)=0$ with zero Neumann data, assuming that the $L^p$ Neumann problem and $L^{p'}$ Dirichlet problem for the adjoint operator are solvable in a Lipschitz cylinder for some $p\in(1,\infty)$. Using this result, we establish the solvability of the Neumann problem in the atomic Hardy space for parabolic operators with bounded, measurable, time-dependent coefficients, and hence obtain the extrapolation of solvability of the $L^p$ Neumann problem. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2601_12429 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Localization and interpolation of parabolic $L^p$ Neumann problems Dindoš, Martin Li, Linhan Pipher, Jill Analysis of PDEs Classical Analysis and ODEs 35K20, 35K10 We show a localization estimate for local solutions to the parabolic equation $-\partial_t u+\mbox{div} (A\nabla u)=0$ with zero Neumann data, assuming that the $L^p$ Neumann problem and $L^{p'}$ Dirichlet problem for the adjoint operator are solvable in a Lipschitz cylinder for some $p\in(1,\infty)$. Using this result, we establish the solvability of the Neumann problem in the atomic Hardy space for parabolic operators with bounded, measurable, time-dependent coefficients, and hence obtain the extrapolation of solvability of the $L^p$ Neumann problem. |
| title | Localization and interpolation of parabolic $L^p$ Neumann problems |
| topic | Analysis of PDEs Classical Analysis and ODEs 35K20, 35K10 |
| url | https://arxiv.org/abs/2601.12429 |