Optimal trace norms for Helmholtz problems
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866915341931642880 |
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| author | Gräßle, Benedikt |
| author_facet | Gräßle, Benedikt |
| contents | The natural $H^1(Ω)$ energy norm for Helmholtz problems is weighted with the wavenumber modulus $σ$ and induces natural weighted norms on the trace spaces $H^{\pm1/2}(Γ)$ by minimial extension to $Ω\subset\mathbb R^n$. This paper presents a rigorous analysis for these trace norms with an explicit characterisation by weighted Sobolev-Slobodeckij norms and scaling estimates, highlighting their dependence on the geometry of the extension set $Ω\subset\mathbb R^n$ and the weight $σ$. The analysis identifies conditions under which these trace norms are intrinsic to the isolated boundary component $Γ\subset\partialΩ$ and provides $σ$-explicit estimates for trace inequalities in weighted spaces. In these natural wavenumber-weighted norms, the boundary integral operators allow improved continuity estimates that do \emph{not} deterioriate as $σ\to 0$. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2506_11944 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Optimal trace norms for Helmholtz problems Gräßle, Benedikt Analysis of PDEs Numerical Analysis 46E35, 35J05, 42B37 The natural $H^1(Ω)$ energy norm for Helmholtz problems is weighted with the wavenumber modulus $σ$ and induces natural weighted norms on the trace spaces $H^{\pm1/2}(Γ)$ by minimial extension to $Ω\subset\mathbb R^n$. This paper presents a rigorous analysis for these trace norms with an explicit characterisation by weighted Sobolev-Slobodeckij norms and scaling estimates, highlighting their dependence on the geometry of the extension set $Ω\subset\mathbb R^n$ and the weight $σ$. The analysis identifies conditions under which these trace norms are intrinsic to the isolated boundary component $Γ\subset\partialΩ$ and provides $σ$-explicit estimates for trace inequalities in weighted spaces. In these natural wavenumber-weighted norms, the boundary integral operators allow improved continuity estimates that do \emph{not} deterioriate as $σ\to 0$. |
| title | Optimal trace norms for Helmholtz problems |
| topic | Analysis of PDEs Numerical Analysis 46E35, 35J05, 42B37 |
| url | https://arxiv.org/abs/2506.11944 |