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

TL;DR

This work addresses the elastic Calderón problem of recovering a spatial density ρ from boundary Neumann-to-Dirichlet data for the time-harmonic Lamé system. By embedding a subwavelength periodic cluster of resonant hard inclusions, the authors create an effective medium with a uniform negative density shift, enabling a near-linear relationship between the N--D map and the unknown density around a negative background. They establish a quantitative convergence Λ_D → Λ_P, derive a first-order linearisation around the background, and, using complex geometric optics solutions, obtain an explicit Fourier-reconstruction formula for ρ with controlled error. The results provide a metamaterial-inspired, analytically tractable framework for elastic inverse problems and a concrete approach for density reconstruction via nanoscale resonators.

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 $\|Λ_D-Λ_{\mathcal{P}}\|\le Ca^α\mathcal{P}^6$ 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.

Figures (2)

• Figure 1: An illustration of how the hard inclusion are distributed in $\Omega$.
• Figure 2: Flow Chart of the proof strategy of Theorem \ref{['thm:N--D']}

Theorems & Definitions (29)

• Theorem 1.1
• Remark 1.1
• Theorem 1.2
• Theorem 1.3
• Remark 1.2
• Remark 1.3
• Lemma 2.1
• proof
• Lemma 2.2
• proof