High-Contrast Transmission Resonances for the Lam\'e System

Abstract

We consider the Lam\'e transmission problem in $\mathbb{R}^3$ with a bounded isotropic elastic inclusion in a high-contrast setting, where the interior-to-exterior Lam\'e moduli and densities scale like $1/\tau$ as $\tau\to0$. 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 $\tau\to0$. Near each nonzero Neumann eigenvalue of the interior Lam\'e 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 $\tau$ with non-vanishing leading coefficients. In addition, near zero (a subwavelength regime), we identify resonances with real parts of order $\sqrt{\tau}$ and prove a lifetime dichotomy: their imaginary parts are of order $\tau$ generically, but of order $\tau^2$ 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.