Sub-wavelength resonances in two-dimensional multi-layer elastic media

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 ${\widehat{\mathbf{S}}}_{\mathrm{\partial }D}^{\omega }$, which serves as a leading order approximation to ${\mathbf{S}}_{\mathrm{\partial }D}^{\omega }$ as $\omega \to 0$, is invertible in the space $\mathcal{L}(L^2(\mathrm{\partial }D{)}^2,H^1(\mathrm{\partial }D{)}^2)$. 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 $\mathcal{O}({\omega }^{-2})$ as the incident frequency $\omega$ 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.