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| Format: | Preprint |
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2026
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| Online Access: | https://arxiv.org/abs/2601.10290 |
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| _version_ | 1866908768675037184 |
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| author | Li, Long Sini, Mourad |
| author_facet | Li, Long Sini, Mourad |
| contents | We consider the Lamé transmission problem in $\mathbb{R}^3$ with a bounded isotropic elastic inclusion in a high-contrast setting, where the interior-to-exterior Lamé moduli and densities scale like $1/τ$ as $τ\to0$. We study the scattering resonances of the associated self-adjoint Hamiltonian, defined as the poles of the meromorphic continuation of its resolvent.
We obtain a sharp asymptotic description of resonances near the real axis as $τ\to0$. Near each nonzero Neumann eigenvalue of the interior Lamé operator there is a cluster of resonances lying just below it in the complex plane; in this wavelength-scale regime the imaginary parts are of order $τ$ with non-vanishing leading coefficients. In addition, near zero (a subwavelength regime), we identify resonances with real parts of order $\sqrtτ$ and prove a lifetime dichotomy: their imaginary parts are of order $τ$ generically, but of order $τ^2$ for an explicit admissible set $\mathcal E$. This yields a classification of long-lived elastic resonances in the high-contrast limit.
We also establish resolvent asymptotics for both fixed-size resonators and microresonators. We derive explicit expansions with a finite-rank leading term and quantitative remainder bounds, valid near both wavelength-scale and subwavelength resonances. For microresonators, at the wavelength scale the dominant contribution is an anisotropic elastic point scatterer. Near the zero eigenvalue, the leading-order behaviour is of monopole or dipole type, and we give a rigorous criterion distinguishing the two cases. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2601_10290 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | High-Contrast Transmission Resonances for the Lamé System Li, Long Sini, Mourad Analysis of PDEs We consider the Lamé transmission problem in $\mathbb{R}^3$ with a bounded isotropic elastic inclusion in a high-contrast setting, where the interior-to-exterior Lamé moduli and densities scale like $1/τ$ as $τ\to0$. We study the scattering resonances of the associated self-adjoint Hamiltonian, defined as the poles of the meromorphic continuation of its resolvent. We obtain a sharp asymptotic description of resonances near the real axis as $τ\to0$. Near each nonzero Neumann eigenvalue of the interior Lamé operator there is a cluster of resonances lying just below it in the complex plane; in this wavelength-scale regime the imaginary parts are of order $τ$ with non-vanishing leading coefficients. In addition, near zero (a subwavelength regime), we identify resonances with real parts of order $\sqrtτ$ and prove a lifetime dichotomy: their imaginary parts are of order $τ$ generically, but of order $τ^2$ for an explicit admissible set $\mathcal E$. This yields a classification of long-lived elastic resonances in the high-contrast limit. We also establish resolvent asymptotics for both fixed-size resonators and microresonators. We derive explicit expansions with a finite-rank leading term and quantitative remainder bounds, valid near both wavelength-scale and subwavelength resonances. For microresonators, at the wavelength scale the dominant contribution is an anisotropic elastic point scatterer. Near the zero eigenvalue, the leading-order behaviour is of monopole or dipole type, and we give a rigorous criterion distinguishing the two cases. |
| title | High-Contrast Transmission Resonances for the Lamé System |
| topic | Analysis of PDEs |
| url | https://arxiv.org/abs/2601.10290 |