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| Main Authors: | , , , |
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| Format: | Preprint |
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2025
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2503.00873 |
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| _version_ | 1866917942911827968 |
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| author | Bortz, Simon Ferris, Sandra Hidalgo-Palencia, Pablo Hofmann, Steve |
| author_facet | Bortz, Simon Ferris, Sandra Hidalgo-Palencia, Pablo Hofmann, Steve |
| contents | We investigate variable coefficient analogs of a recent work of Bortz, Hofmann, Martell and Nyström [BHMN25]. In particular, we show that if $Ω$ is the region above the graph of a Lip(1,1/2) (parabolic Lipschitz) function and $L$ is a parabolic operator in divergence form \[L = \partial_t - \text{div} A \nabla\] with $A$ satisfying an $L^1$ Carleson condition on its spatial and time derivatives, then the $L^p$-solvability of the Dirichlet problem for $L$ and $L^*$ implies that the graph function has a half-order time derivative in BMO. Equivalently, the graph is parabolic uniformly rectifiable.
In the case of $A$ symmetric, we only require that the Dirichlet problem for $L$ is solvable, which requires us to adapt a clever integration by parts argument by Lewis and Nyström. A feature of the present work is that we must overcome the lack of translation invariance in our equation, which is a fundamental tool in similar works, including [BHMN25]. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2503_00873 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | A Variable Coefficient Free Boundary Problem for $L^p$-solvability of Parabolic Dirichlet Problems in Graph Domains Bortz, Simon Ferris, Sandra Hidalgo-Palencia, Pablo Hofmann, Steve Analysis of PDEs Classical Analysis and ODEs We investigate variable coefficient analogs of a recent work of Bortz, Hofmann, Martell and Nyström [BHMN25]. In particular, we show that if $Ω$ is the region above the graph of a Lip(1,1/2) (parabolic Lipschitz) function and $L$ is a parabolic operator in divergence form \[L = \partial_t - \text{div} A \nabla\] with $A$ satisfying an $L^1$ Carleson condition on its spatial and time derivatives, then the $L^p$-solvability of the Dirichlet problem for $L$ and $L^*$ implies that the graph function has a half-order time derivative in BMO. Equivalently, the graph is parabolic uniformly rectifiable. In the case of $A$ symmetric, we only require that the Dirichlet problem for $L$ is solvable, which requires us to adapt a clever integration by parts argument by Lewis and Nyström. A feature of the present work is that we must overcome the lack of translation invariance in our equation, which is a fundamental tool in similar works, including [BHMN25]. |
| title | A Variable Coefficient Free Boundary Problem for $L^p$-solvability of Parabolic Dirichlet Problems in Graph Domains |
| topic | Analysis of PDEs Classical Analysis and ODEs |
| url | https://arxiv.org/abs/2503.00873 |