A nonvariational form of the acoustic single layer potential
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arXiv
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| Format: | Preprint |
| Publié: |
2025
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| _version_ | 1866912802265890816 |
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| author | de Cristoforis, M. Lanza |
| author_facet | de Cristoforis, M. Lanza |
| contents | We consider a bounded open subset $Ω$ of ${\mathbb{R}}^n$ of class $C^{1,α}$ for some $α\in]0,1[$ and the space $V^{-1,α}(\partialΩ)$ of (distributional) normal derivatives on the boundary of $α$-Hölder continuous functions in $Ω$ that have Laplace operator in the Schauder space with negative exponent $C^{-1,α}(\overlineΩ)$. Then we prove those properties of the acoustic single layer potential that are necessary to analyze the Neumann problem for the Helmholtz equation in $Ω$ with boundary data in $V^{-1,α}(\partialΩ)$ and solutions in the space of $α$-Hölder continuous functions in $Ω$ that have Laplace operator in $C^{-1,α}(\overlineΩ)$, \textit{i.e.}, in a space of functions that may have infinite Dirichlet integral. Namely, a Neumann problem that does not belong to the classical variational setting. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2504_12349 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | A nonvariational form of the acoustic single layer potential de Cristoforis, M. Lanza Analysis of PDEs 31B10, 35J25, 35J05 We consider a bounded open subset $Ω$ of ${\mathbb{R}}^n$ of class $C^{1,α}$ for some $α\in]0,1[$ and the space $V^{-1,α}(\partialΩ)$ of (distributional) normal derivatives on the boundary of $α$-Hölder continuous functions in $Ω$ that have Laplace operator in the Schauder space with negative exponent $C^{-1,α}(\overlineΩ)$. Then we prove those properties of the acoustic single layer potential that are necessary to analyze the Neumann problem for the Helmholtz equation in $Ω$ with boundary data in $V^{-1,α}(\partialΩ)$ and solutions in the space of $α$-Hölder continuous functions in $Ω$ that have Laplace operator in $C^{-1,α}(\overlineΩ)$, \textit{i.e.}, in a space of functions that may have infinite Dirichlet integral. Namely, a Neumann problem that does not belong to the classical variational setting. |
| title | A nonvariational form of the acoustic single layer potential |
| topic | Analysis of PDEs 31B10, 35J25, 35J05 |
| url | https://arxiv.org/abs/2504.12349 |