Elastic Calderón Problem via Resonant Hard Inclusions: Linearisation of the N-D Map and Density Reconstruction

Abstract: We study an elastic Calderon-type inverse problem: recover the mass density $ρ(x)$ in a bounded domain $Ω\subset\mathbb{R}^3$ from the Neumann-to-Dirichlet map associated with the isotropic Lamé system $\mathcal{L}{λ,μ}u+ω^2ρ(x)u=0$. 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 $ρ_1\asymp a^{-2}$ strictly inside $Ω$. For frequencies $ω$ 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 $Λ_D$ converges to an effective map $Λ{\mathcal{P}}$ corresponding to a background density shift $-\mathcal{P}^2$, with the operator norm estimate $|ΛD-Λ{\mathcal{P}}|\le Ca^α\mathcal{P}^6$ for some $α>0$ determined by the geometric scaling. Around this negative background we derive a first-order linearization of $Λ_{\mathcal{P}}$ in terms of $ρ$ and the Newton volume potential for the shifted Lamé operator. Testing the linearized relation with complex geometric optics solutions yields an explicit reconstruction formula for the Fourier transform of $ρ$, 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.