High-Contrast Transmission Resonances for the Lamé System

TL;DR

The paper analyzes the high-contrast Lamé transmission problem in $\mathbb{R}^3$ with an isotropic inclusion whose interior parameters scale like $1/\tau$ as $\tau\to 0$. It develops a rigorous boundary-integral and finite-dimensional reduction framework yielding sharp asymptotics for scattering resonances: near nonzero Neumann eigenfrequencies, resonances cluster just below $z_0$ with widths $\mathcal{O}(\tau)$ determined by eigenvalues of an explicit effective matrix $M^{(1)}(z_0)$, and near zero, subwavelength resonances have real parts $\sim \sqrt{\tau}$ with a lifetime exhibiting a dichotomy governed by an admissible set $\mathcal{E}$. The work also derives resolvent expansions for both fixed-size resonators and microresonators, revealing finite-rank leading terms: a resonant interior expansion coupled to an exterior radiating field at the wavelength scale, and monopole- or dipole-like point-scatterer descriptions at the subwavelength scale. The results provide a rigorous mechanism for long-lived elastic resonances and anisotropic subwavelength scattering, with direct implications for designing elastic metamaterials and effective medium theories. The analysis extends prior scalar high-contrast results to the vectorial Lamé system, illustrating a rich interaction between longitudinal and shear modes through a vector-valued enhancement-space framework and explicit perturbative formulas for resonance widths.

Abstract

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.

Figures (2)

• Figure 3.1: Schematic location of scattering resonances in the high-contrast limit $\tau\to0$. Theorem \ref{['th:0']} yields that resonances are confined to small disks around the interior Neumann spectrum and that a resonance-free domain persists away from these disks. For each nonzero Neumann eigenvalue $z_0\in\Lambda\setminus\{0\}$, Theorem \ref{['th:1']} shows that a finite cluster of resonances lies just below $z_0$, with $\mathrm{Im}(z^{(l)}(\tau))\sim-\tau$. Near $0$, Theorem \ref{['th:2']} yields subwavelength resonances with $|\mathrm{Re}(z(\tau))|\sim \sqrt{\tau}$ and $\mathrm{Im}(z(\tau))\sim -\tau$ generically, while the exceptional $\tau^2$-regime may also occur.
• Figure 3.2: Role of the admissible set $\mathcal{E}$ in the subwavelength regime (Theorem \ref{['th:2']}). Left: for fixed small $\tau$, both regimes have $\mathrm{Re}(z(\tau)) \sim \pm \sqrt{\tau}$, but the depth below the real axis differs. Right: log scaling of $|\mathrm{Im}(z(\tau))|$: generically $|\mathrm{Im}{(z(\tau))}|\sim \tau$ (leading to lifetime $T\sim \tau^{-1}$), while if $\mathcal{E}\neq\emptyset$ there exist resonances with $|\mathrm{Im}( z(\tau))|\sim \tau^2$ (hence $T\sim \tau^{-2}$).

Theorems & Definitions (48)

• Remark 2.1: Nonemptiness of the admissible set $\mathcal{E}$
• Remark 2.2: Interpretation of the admissible set $\mathcal{E}$
• Theorem 3.1
• Theorem 3.2
• Theorem 3.3
• Theorem 3.4
• Theorem 3.5
• Theorem 3.6
• Theorem 3.7
• Proposition 4.1