On the regularity problem for parabolic operators and the role of half-time derivative
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arXiv
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| Natura: | Preprint |
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2023
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| _version_ | 1866908275607339008 |
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| author | Dindoš, Martin |
| author_facet | Dindoš, Martin |
| contents | In this paper we present the following result on regularity of solutions of the second order parabolic equation $\partial_t u - \mbox{div} (A \nabla u)+B\cdot \nabla u=0$ on cylindrical domains of the form $Ω=\mathcal O\times\mathbb R$ where $\mathcal O\subset\mathbb R^n$ is a is a uniform domain (it satisfies both interior corkscrew and Harnack chain conditions) and has a boundary that is $n-1$-Ahlfors regular. Let $u$ be a solution of such PDE in $Ω$ and the non-tangential maximal function of its gradient in spatial directions $\tilde{N}(\nabla u)$ belongs to $L^p(\partialΩ)$ for some $p>1$. Furthermore, assume that for $u|_{\partialΩ}=f$ we have that $D^{1/2}_tf\in L^p(\partialΩ)$. Then both $\tilde{N}(D^{1/2}_t u)$ and $\tilde{N}(D^{1/2}_tH_t u)$ also belong to $L^p(\partialΩ)$, where $D^{1/2}_t$ and $H_t$ are the half-derivative and the Hilbert transform in the time variable, respectively. We expect this result will spur new developments in the study of solvability of the $L^p$ parabolic Regularity problem as thanks to it it is now possible to formulate the parabolic Regularity problem on a large class of time-varying domains. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2308_12936 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | On the regularity problem for parabolic operators and the role of half-time derivative Dindoš, Martin Analysis of PDEs 35K10, 35K20, 35K40, 35K51 In this paper we present the following result on regularity of solutions of the second order parabolic equation $\partial_t u - \mbox{div} (A \nabla u)+B\cdot \nabla u=0$ on cylindrical domains of the form $Ω=\mathcal O\times\mathbb R$ where $\mathcal O\subset\mathbb R^n$ is a is a uniform domain (it satisfies both interior corkscrew and Harnack chain conditions) and has a boundary that is $n-1$-Ahlfors regular. Let $u$ be a solution of such PDE in $Ω$ and the non-tangential maximal function of its gradient in spatial directions $\tilde{N}(\nabla u)$ belongs to $L^p(\partialΩ)$ for some $p>1$. Furthermore, assume that for $u|_{\partialΩ}=f$ we have that $D^{1/2}_tf\in L^p(\partialΩ)$. Then both $\tilde{N}(D^{1/2}_t u)$ and $\tilde{N}(D^{1/2}_tH_t u)$ also belong to $L^p(\partialΩ)$, where $D^{1/2}_t$ and $H_t$ are the half-derivative and the Hilbert transform in the time variable, respectively. We expect this result will spur new developments in the study of solvability of the $L^p$ parabolic Regularity problem as thanks to it it is now possible to formulate the parabolic Regularity problem on a large class of time-varying domains. |
| title | On the regularity problem for parabolic operators and the role of half-time derivative |
| topic | Analysis of PDEs 35K10, 35K20, 35K40, 35K51 |
| url | https://arxiv.org/abs/2308.12936 |