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| Main Authors: | , , , |
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| Format: | Preprint |
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2026
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| Online Access: | https://arxiv.org/abs/2601.12821 |
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| _version_ | 1866912832049643520 |
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| author | Jiang, Yan Liu, Hongyu Sun, Fanbo Wang, Yajuan |
| author_facet | Jiang, Yan Liu, Hongyu Sun, Fanbo Wang, Yajuan |
| contents | In this paper, we focus on the sub-wavelength resonances in two-dimensional elastic media characterized by high contrasts in both Lamé parameters and density. Our contributions are fourfold. First, it is proved that the operator $\hat{\mathbf{S}}_{\partial D}^ω$, which serves as a leading order approximation to $\mathbf{S}_{\partial D}^ω$ as $ω\rightarrow0$, is invertible in the space $\mathcal{L}(L^{2}\left(\partial D)^{2},H^{1}(\partial D)^{2}\right)$. Second, based on layer potential techniques in combination with asymptotic analysis, we derive an original formula for the leading-order terms of sub-wavelength resonance frequencies, which are controlled by the determinant of the $3N \times 3N$ matrices. Specifically, there are $3N$ resonance frequencies within an $N$-nested layer structure. In addition, the scattering field exhibits an enhancement coefficient on the order of $\mathcal{O}(ω^{-2})$ as the incident frequency $ω$ approaches the resonance frequency. Third, by applying spectral properties to solve the corresponding eigenvalue problem, we compute the quantitative expressions for sub-wavelength resonance frequencies within a disk. Finally, some numerical experiments are provided to illustrate theoretical results and demonstrate the existence of the sub-wavelength resonance modes. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2601_12821 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Sub-wavelength resonances in two-dimensional multi-layer elastic media Jiang, Yan Liu, Hongyu Sun, Fanbo Wang, Yajuan Analysis of PDEs 35R30, 35B30, 35B34 In this paper, we focus on the sub-wavelength resonances in two-dimensional elastic media characterized by high contrasts in both Lamé parameters and density. Our contributions are fourfold. First, it is proved that the operator $\hat{\mathbf{S}}_{\partial D}^ω$, which serves as a leading order approximation to $\mathbf{S}_{\partial D}^ω$ as $ω\rightarrow0$, is invertible in the space $\mathcal{L}(L^{2}\left(\partial D)^{2},H^{1}(\partial D)^{2}\right)$. Second, based on layer potential techniques in combination with asymptotic analysis, we derive an original formula for the leading-order terms of sub-wavelength resonance frequencies, which are controlled by the determinant of the $3N \times 3N$ matrices. Specifically, there are $3N$ resonance frequencies within an $N$-nested layer structure. In addition, the scattering field exhibits an enhancement coefficient on the order of $\mathcal{O}(ω^{-2})$ as the incident frequency $ω$ approaches the resonance frequency. Third, by applying spectral properties to solve the corresponding eigenvalue problem, we compute the quantitative expressions for sub-wavelength resonance frequencies within a disk. Finally, some numerical experiments are provided to illustrate theoretical results and demonstrate the existence of the sub-wavelength resonance modes. |
| title | Sub-wavelength resonances in two-dimensional multi-layer elastic media |
| topic | Analysis of PDEs 35R30, 35B30, 35B34 |
| url | https://arxiv.org/abs/2601.12821 |