A uniqueness theorem for nonvariational solutions of the Helmholtz equation
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866912329976774656 |
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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 we define a distributional outward unit normal derivative for $α$-Hölder continuous solutions of the Helmholtz equation in the exterior of $Ω$ that may not have a classical outward unit normal derivative at the boundary points of $Ω$ and that may have an infinite Dirichlet integral around the boundary of $Ω$. Namely for solutions that do not belong to the classical variational setting. Then we show a Schauder boundary regularity result for $α$-Hölder continuous functions that have the Laplace operator in a Schauder space of negative exponent and we prove a uniqueness theorem for $α$-Hölder continuous solutions of the exterior Dirichlet and impedance boundary value problems for the Helmholtz equation that satisfy the Sommerfeld radiation condition at infinity in the above mentioned nonvariational setting. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2504_11487 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | A uniqueness theorem for nonvariational solutions of the Helmholtz equation 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 we define a distributional outward unit normal derivative for $α$-Hölder continuous solutions of the Helmholtz equation in the exterior of $Ω$ that may not have a classical outward unit normal derivative at the boundary points of $Ω$ and that may have an infinite Dirichlet integral around the boundary of $Ω$. Namely for solutions that do not belong to the classical variational setting. Then we show a Schauder boundary regularity result for $α$-Hölder continuous functions that have the Laplace operator in a Schauder space of negative exponent and we prove a uniqueness theorem for $α$-Hölder continuous solutions of the exterior Dirichlet and impedance boundary value problems for the Helmholtz equation that satisfy the Sommerfeld radiation condition at infinity in the above mentioned nonvariational setting. |
| title | A uniqueness theorem for nonvariational solutions of the Helmholtz equation |
| topic | Analysis of PDEs 31B10, 35J25, 35J05 |
| url | https://arxiv.org/abs/2504.11487 |