Elastic Calder\'on Problem via Resonant Hard Inclusions: Linearisation of the N-D Map and Density Reconstruction

We study an elastic Calderon-type inverse problem: recover the mass density $\rho(x)$ in a bounded domain $\Omega\subset\mathbb{R}^3$ from the Neumann-to-Dirichlet map associated with the isotropic Lam\'e system $\mathcal{L}_{\lambda,\mu}u+\omega^2\rho(x)u=0$. We introduce a constructive strategy that embeds a subwavelength periodic array of resonant high-density (hard) inclusions to create an effective medium with a uniform negative density shift. Specifically, we place a periodic cluster of inclusions of size $a$ and density $\rho_1\asymp a^{-2}$ strictly inside $\Omega$. For frequencies $\omega$ tuned to an eigenvalue of the elastic Newton (Kelvin) operator of a single inclusion, we show that as $a\to0$ and the number of inclusions $M\to\infty$, the Neumann-to-Dirichlet map $\Lambda_D$ converges to an effective map $\Lambda_{\mathcal{P}}$ corresponding to a background density shift $-\mathcal{P}^2$, with the operator norm estimate $\|\Lambda_D-\Lambda_{\mathcal{P}}\|\le Ca^{\alpha}\mathcal{P}^6$ for some $\alpha>0$ determined by the geometric scaling. Around this negative background we derive a first-order linearization of $\Lambda_{\mathcal{P}}$ in terms of $\rho$ and the Newton volume potential for the shifted Lam\'e operator. Testing the linearized relation with complex geometric optics solutions yields an explicit reconstruction formula for the Fourier transform of $\rho$, and hence a global density recovery scheme. The results provide a metamaterial-inspired analytic framework for inverse coefficient problems in linear elasticity and a concrete paradigm for leveraging nanoscale resonators in reconstruction algorithms.