Solvability of the $L^p$ Dirichlet problem for the heat equation implies parabolic uniform rectifiability
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arXiv
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| Natura: | Preprint |
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2025
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| _version_ | 1866912669805576192 |
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| author | Bortz, Simon Hofmann, Steven Martell, José María Nyström, Kaj |
| author_facet | Bortz, Simon Hofmann, Steven Martell, José María Nyström, Kaj |
| contents | Let $Ω\subset \mathbb{R}^{n+1}$ be an open set in space-time with boundary $Σ= \partial Ω$. Under minimal and natural background assumptions - namely, that $Σ$ is time-symmetrically parabolic Ahlfors--David regular and that $Ω$ satisfies an interior corkscrew condition - we treat a one-phase parabolic free boundary problem which establishes the necessity of parabolic uniform rectifiability for $L^p(dσ)$ solvability of the Dirichlet problem for the heat equation. More precisely, we prove that if the caloric measure associated with $Ω$ satisfies a weak-$A_\infty$ condition with respect to the surface measure $σ= \mathcal{H}_{\mathrm{par}}^{n+1}\!\lfloor_Σ$, then $Σ$ is parabolically uniformly rectifiable, hence equivalently, that solvability of the Dirichlet problem for the heat (or adjoint heat) equation in $Ω$ with boundary data in $L^p(dσ)$, for some $p \in (1,\infty)$, implies parabolic uniform rectifiability. Our main theorem thus identifies parabolic uniform rectifiability as the correct geometric framework for boundary regularity, and $L^p$ solvability, in the parabolic setting. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2510_22047 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Solvability of the $L^p$ Dirichlet problem for the heat equation implies parabolic uniform rectifiability Bortz, Simon Hofmann, Steven Martell, José María Nyström, Kaj Analysis of PDEs Classical Analysis and ODEs 28A75, 35K05, 35K20, 35R35, 42B25, 42B37, 43A85 Let $Ω\subset \mathbb{R}^{n+1}$ be an open set in space-time with boundary $Σ= \partial Ω$. Under minimal and natural background assumptions - namely, that $Σ$ is time-symmetrically parabolic Ahlfors--David regular and that $Ω$ satisfies an interior corkscrew condition - we treat a one-phase parabolic free boundary problem which establishes the necessity of parabolic uniform rectifiability for $L^p(dσ)$ solvability of the Dirichlet problem for the heat equation. More precisely, we prove that if the caloric measure associated with $Ω$ satisfies a weak-$A_\infty$ condition with respect to the surface measure $σ= \mathcal{H}_{\mathrm{par}}^{n+1}\!\lfloor_Σ$, then $Σ$ is parabolically uniformly rectifiable, hence equivalently, that solvability of the Dirichlet problem for the heat (or adjoint heat) equation in $Ω$ with boundary data in $L^p(dσ)$, for some $p \in (1,\infty)$, implies parabolic uniform rectifiability. Our main theorem thus identifies parabolic uniform rectifiability as the correct geometric framework for boundary regularity, and $L^p$ solvability, in the parabolic setting. |
| title | Solvability of the $L^p$ Dirichlet problem for the heat equation implies parabolic uniform rectifiability |
| topic | Analysis of PDEs Classical Analysis and ODEs 28A75, 35K05, 35K20, 35R35, 42B25, 42B37, 43A85 |
| url | https://arxiv.org/abs/2510.22047 |