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| Autor principal: | |
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| Formato: | Preprint |
| Publicado: |
2024
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| Materias: | |
| Acceso en línea: | https://arxiv.org/abs/2405.04388 |
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| _version_ | 1866918452562755584 |
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| author | Vita, Stefano |
| author_facet | Vita, Stefano |
| contents | Let $Ω\subset\mathbb R^2$ be a chord arc domain. We give a simple proof of the the following fact, which is commonly known to be true: a nontrivial harmonic function which vanishes continuously on a relatively open set of the boundary cannot have the norm of the gradient which vanishes on a subset of positive surface measure (arc length). This result is conjectured to be true in higher dimensions by Lin, in Lipschitz domains. Let now $Ω\subset\mathbb R^2$ be a $C^1$ domain with Dini mean oscillations. We prove that a nontrivial harmonic function which vanishes continuously on a relatively open subset of the boundary $\partialΩ\cap B_1$ has a finite number of critical points in $\overlineΩ\cap B_{1/2}$. The latter improves some recent results by Kenig and Zhao. Our technique involves a conformal mapping which moves the boundary where the harmonic function vanishes into an interior nodal line of a new harmonic function, after a further reflection. Then, size estimates of the critical set - up to the boundary - of the original harmonic function can be understood in terms of estimates of the \emph{interior} critical set of the new harmonic function and of the critical set - up to the boundary - of the conformal mapping. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2405_04388 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Boundary unique continuation in planar domains by conformal mapping Vita, Stefano Analysis of PDEs 30C20, 31A05, 35J25, 42B37 Let $Ω\subset\mathbb R^2$ be a chord arc domain. We give a simple proof of the the following fact, which is commonly known to be true: a nontrivial harmonic function which vanishes continuously on a relatively open set of the boundary cannot have the norm of the gradient which vanishes on a subset of positive surface measure (arc length). This result is conjectured to be true in higher dimensions by Lin, in Lipschitz domains. Let now $Ω\subset\mathbb R^2$ be a $C^1$ domain with Dini mean oscillations. We prove that a nontrivial harmonic function which vanishes continuously on a relatively open subset of the boundary $\partialΩ\cap B_1$ has a finite number of critical points in $\overlineΩ\cap B_{1/2}$. The latter improves some recent results by Kenig and Zhao. Our technique involves a conformal mapping which moves the boundary where the harmonic function vanishes into an interior nodal line of a new harmonic function, after a further reflection. Then, size estimates of the critical set - up to the boundary - of the original harmonic function can be understood in terms of estimates of the \emph{interior} critical set of the new harmonic function and of the critical set - up to the boundary - of the conformal mapping. |
| title | Boundary unique continuation in planar domains by conformal mapping |
| topic | Analysis of PDEs 30C20, 31A05, 35J25, 42B37 |
| url | https://arxiv.org/abs/2405.04388 |