Extrapolation of solvability of the parabolic $L^p$ Neumann problem on bounded Lipschitz cylinders
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arXiv
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| Natura: | Preprint |
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2026
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| _version_ | 1866917353149693952 |
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| author | Dindoš, Martin Liu, YingYi |
| author_facet | Dindoš, Martin Liu, YingYi |
| contents | A recent result of the first author with Li and Pipher has established the extrapolation of solvability of the $L^p$ parabolic Neumann problem on unbounded graph domains of the form $Ω=\{(x',x_n):\,x_n>φ(x')\}\times\mathbb R$, where $φ:\mathbb R^{n-1}\to\mathbb R$ is a Lipschitz function. The result shows that under the assumptions that the $L^p$ parabolic Neumann problem for the equation $Lu=-\partial_t u+\mbox{div}(A\nabla u)=0$ in $Ω$ and also the $L^{p'}$ parabolic Dirichlet problem for the adjoint equation $L^*u=\partial_t u+\mbox{div}(A\nabla u)=0$ in $Ω$ are solvable, then also the $L^q$ parabolic Neumann problem for the equation $Lu=0$ in $Ω$ is solvable for all $1<q<p$.
However the mentioned paper does not answer the question whether the same claim is also true for domains of the form $\mathcal O\times\mathbb R$, where $\mathcal O$ is a bounded Lipschitz domain (in spatial variables) since this case does not follow from our argument for the unbounded case. Indeed, the bounded Lipschitz cylinder case requires a significantly different approach which we present in this article and establish an analogous result when $\mathcal O$ is a bounded Lipschitz domain. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2603_15898 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Extrapolation of solvability of the parabolic $L^p$ Neumann problem on bounded Lipschitz cylinders Dindoš, Martin Liu, YingYi Analysis of PDEs Classical Analysis and ODEs 35K20, 35K10 A recent result of the first author with Li and Pipher has established the extrapolation of solvability of the $L^p$ parabolic Neumann problem on unbounded graph domains of the form $Ω=\{(x',x_n):\,x_n>φ(x')\}\times\mathbb R$, where $φ:\mathbb R^{n-1}\to\mathbb R$ is a Lipschitz function. The result shows that under the assumptions that the $L^p$ parabolic Neumann problem for the equation $Lu=-\partial_t u+\mbox{div}(A\nabla u)=0$ in $Ω$ and also the $L^{p'}$ parabolic Dirichlet problem for the adjoint equation $L^*u=\partial_t u+\mbox{div}(A\nabla u)=0$ in $Ω$ are solvable, then also the $L^q$ parabolic Neumann problem for the equation $Lu=0$ in $Ω$ is solvable for all $1<q<p$. However the mentioned paper does not answer the question whether the same claim is also true for domains of the form $\mathcal O\times\mathbb R$, where $\mathcal O$ is a bounded Lipschitz domain (in spatial variables) since this case does not follow from our argument for the unbounded case. Indeed, the bounded Lipschitz cylinder case requires a significantly different approach which we present in this article and establish an analogous result when $\mathcal O$ is a bounded Lipschitz domain. |
| title | Extrapolation of solvability of the parabolic $L^p$ Neumann problem on bounded Lipschitz cylinders |
| topic | Analysis of PDEs Classical Analysis and ODEs 35K20, 35K10 |
| url | https://arxiv.org/abs/2603.15898 |