The $L^p$ regularity problem for parabolic operators
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| Format: | Preprint |
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2024
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| _version_ | 1866913060137992192 |
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| author | Dindoš, Martin Li, Linhan Pipher, Jill |
| author_facet | Dindoš, Martin Li, Linhan Pipher, Jill |
| contents | In this paper, we fully resolve the question of whether the Regularity problem for the parabolic PDE $-\partial_tu + \mbox{div}(A\nabla u)=0$ on a Lipschitz cylinder $\mathcal O\times\mathbb R$ is solvable for some $p\in (1,\infty)$ under the assumption that the matrix $A$ is elliptic, has bounded and measurable coefficients and its coefficients satisfy a natural Carleson condition (a parabolic analog of the so-called DKP-condition).
We prove that for some $p_0>1$ the Regularity problem is solvable in the range $(1,p_0)$. We note that answer to this question was not known even in the small Carleson case, that is, when the Carleson norm of coefficients is sufficiently small.
In the elliptic case the analogous question was only fully resolved recently independently by two groups, with two very different methods: one involving two of the authors and S. Hofmann, the second by M. Mourgoglou, B. Poggi and X. Tolsa. Our approach in the parabolic case is motivated by that of the first group, but in the parabolic setting there are significant new challenges. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2410_23801 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | The $L^p$ regularity problem for parabolic operators Dindoš, Martin Li, Linhan Pipher, Jill Analysis of PDEs Classical Analysis and ODEs 35K20, 35K10 In this paper, we fully resolve the question of whether the Regularity problem for the parabolic PDE $-\partial_tu + \mbox{div}(A\nabla u)=0$ on a Lipschitz cylinder $\mathcal O\times\mathbb R$ is solvable for some $p\in (1,\infty)$ under the assumption that the matrix $A$ is elliptic, has bounded and measurable coefficients and its coefficients satisfy a natural Carleson condition (a parabolic analog of the so-called DKP-condition). We prove that for some $p_0>1$ the Regularity problem is solvable in the range $(1,p_0)$. We note that answer to this question was not known even in the small Carleson case, that is, when the Carleson norm of coefficients is sufficiently small. In the elliptic case the analogous question was only fully resolved recently independently by two groups, with two very different methods: one involving two of the authors and S. Hofmann, the second by M. Mourgoglou, B. Poggi and X. Tolsa. Our approach in the parabolic case is motivated by that of the first group, but in the parabolic setting there are significant new challenges. |
| title | The $L^p$ regularity problem for parabolic operators |
| topic | Analysis of PDEs Classical Analysis and ODEs 35K20, 35K10 |
| url | https://arxiv.org/abs/2410.23801 |