Elastic Calderón-Type Inverse Problem

• The paper introduces a metamaterial-inspired approach that uses high-density hard inclusions to achieve an effective negative density shift, thereby linearizing the otherwise nonlinear elastic inverse problem.
• It employs complex geometric optics solutions to relate boundary measurement differences to the Fourier samples of the unknown mass density, enabling explicit density recovery.
• The method integrates homogenization techniques with resonant microstructure design, bridging advanced mathematical theory with practical reconstruction algorithms in elasticity.

An elastic Calderón-type inverse problem refers to the determination of an unknown mass density $\rho(x)$ within a bounded elastic domain $\Omega \subset \mathbb{R}^3$ using boundary measurements, specifically the Neumann-to-Dirichlet (N–D) map associated to the isotropic Lamé system at fixed frequency. Recent advances leverage metamaterial-inspired strategies, embedding subwavelength clusters of high-density inclusions to induce an effective negative density shift and facilitate analytic linearization of the inverse problem. This approach leads to a global reconstruction algorithm for $\rho(x)$ based on a first-order expansion of the homogenized N–D map and the systematic use of complex geometric optics (CGO) solutions. The following entry provides a comprehensive account of this analytic and constructive framework.

1. Mathematical Formulation of the Elastic Calderón Problem

Let $\Omega \subset \mathbb{R}^3$ be a bounded Lipschitz domain. The forward elasticity problem seeks the displacement $u(x) : \Omega \to \mathbb{C}^3$ solving

$$\begin{cases} \mathcal{L}_{\lambda, \mu} \, u(x) + \omega^2 \rho(x) u(x) = 0, & x \in \Omega \ \partial_{\nu} u(x) = g(x), & x \in \partial\Omega \end{cases}$$

where

$$\mathcal{L}_{\lambda,\mu} u = \mu \Delta u + (\lambda+\mu) \nabla(\nabla\cdot u), \quad \partial_\nu u = [\lambda (\nabla\cdot u) I + 2\mu \nabla^s u]\nu,$$

with Lamé parameters $\lambda, \mu > 0$ and mass density $\rho(x) > 0$. Traction $g$ prescribes the Neumann data on $\Omega \subset \mathbb{R}^3$0. The associated Neumann-to-Dirichlet map $\Omega \subset \mathbb{R}^3$1 encodes the mapping from applied boundary tractions to measured boundary displacements.

The elastic Calderón-type inverse problem is to reconstruct $\Omega \subset \mathbb{R}^3$2 from knowledge of $\Omega \subset \mathbb{R}^3$3 at fixed frequency $\Omega \subset \mathbb{R}^3$4. Without further structure, this problem is both nonlinear and ill-posed.

2. Resonant Hard-Inclusion Array and Homogenization

To address the inherent nonlinearity, the method of Diao, Sini, and Tang introduces a subwavelength periodic array of $\Omega \subset \mathbb{R}^3$5 high-density, "hard" inclusions within a subregion of $\Omega \subset \mathbb{R}^3$6. Each inclusion $\Omega \subset \mathbb{R}^3$7 is a scaled copy (radius $\Omega \subset \mathbb{R}^3$8) of a reference shape $\Omega \subset \mathbb{R}^3$9, located at $\rho(x)$0. The density of inclusions is amplified as $\rho(x)$1 with $\rho(x)$2.

Let $\rho(x)$3 denote the Neumann–Newton (Kelvin) operator on $\rho(x)$4, with eigenpairs $\rho(x)$5. By tuning the driving frequency $\rho(x)$6 near the $\rho(x)$7th eigen-resonance with

$\rho(x)$8

the entire array behaves, in the homogenization limit ($\rho(x)$9, $\Omega \subset \mathbb{R}^3$0), as an effective elastic medium with a uniform negative density shift $\Omega \subset \mathbb{R}^3$1: $\Omega \subset \mathbb{R}^3$2 The perturbed N–D map $\Omega \subset \mathbb{R}^3$3 is shown to converge, in operator norm, to the homogenized map $\Omega \subset \mathbb{R}^3$4,

$\Omega \subset \mathbb{R}^3$5

where $\Omega \subset \mathbb{R}^3$6 is arbitrary.

3. First-Order Linearization Around the Negative Background

For the effective problem with negative density shift, one considers

$\Omega \subset \mathbb{R}^3$7

Given fixed $\Omega \subset \mathbb{R}^3$8, let $\Omega \subset \mathbb{R}^3$9 solve $u(x) : \Omega \to \mathbb{C}^3$0, $u(x) : \Omega \to \mathbb{C}^3$1.

Define the Newtonian volume potential for the shifted operator as

$u(x) : \Omega \to \mathbb{C}^3$2

where $u(x) : \Omega \to \mathbb{C}^3$3 satisfies $u(x) : \Omega \to \mathbb{C}^3$4 and $u(x) : \Omega \to \mathbb{C}^3$5.

A first-order linearization of $u(x) : \Omega \to \mathbb{C}^3$6 in terms of $u(x) : \Omega \to \mathbb{C}^3$7 is established: $u(x) : \Omega \to \mathbb{C}^3$8 with $u(x) : \Omega \to \mathbb{C}^3$9 denoting trace on $$\begin{cases} \mathcal{L}_{\lambda, \mu} \, u(x) + \omega^2 \rho(x) u(x) = 0, & x \in \Omega \ \partial_{\nu} u(x) = g(x), & x \in \partial\Omega \end{cases}$$0. The linearized map is

$$\begin{cases} \mathcal{L}_{\lambda, \mu} \, u(x) + \omega^2 \rho(x) u(x) = 0, & x \in \Omega \ \partial_{\nu} u(x) = g(x), & x \in \partial\Omega \end{cases}$$1

4. Density Recovery Using Complex Geometric Optics (CGO) Solutions

To exploit the linearization, one selects boundary data $$\begin{cases} \mathcal{L}_{\lambda, \mu} \, u(x) + \omega^2 \rho(x) u(x) = 0, & x \in \Omega \ \partial_{\nu} u(x) = g(x), & x \in \partial\Omega \end{cases}$$2 producing internal fields $$\begin{cases} \mathcal{L}_{\lambda, \mu} \, u(x) + \omega^2 \rho(x) u(x) = 0, & x \in \Omega \ \partial_{\nu} u(x) = g(x), & x \in \partial\Omega \end{cases}$$3 that approximate CGO solutions of the form

$$\begin{cases} \mathcal{L}_{\lambda, \mu} \, u(x) + \omega^2 \rho(x) u(x) = 0, & x \in \Omega \ \partial_{\nu} u(x) = g(x), & x \in \partial\Omega \end{cases}$$4

where $$\begin{cases} \mathcal{L}_{\lambda, \mu} \, u(x) + \omega^2 \rho(x) u(x) = 0, & x \in \Omega \ \partial_{\nu} u(x) = g(x), & x \in \partial\Omega \end{cases}$$5, $$\begin{cases} \mathcal{L}_{\lambda, \mu} \, u(x) + \omega^2 \rho(x) u(x) = 0, & x \in \Omega \ \partial_{\nu} u(x) = g(x), & x \in \partial\Omega \end{cases}$$6, and $$\begin{cases} \mathcal{L}_{\lambda, \mu} \, u(x) + \omega^2 \rho(x) u(x) = 0, & x \in \Omega \ \partial_{\nu} u(x) = g(x), & x \in \partial\Omega \end{cases}$$7 is large; $$\begin{cases} \mathcal{L}_{\lambda, \mu} \, u(x) + \omega^2 \rho(x) u(x) = 0, & x \in \Omega \ \partial_{\nu} u(x) = g(x), & x \in \partial\Omega \end{cases}$$8 are lower-order corrections. This ansatz yields asymptotically

$$\begin{cases} \mathcal{L}_{\lambda, \mu} \, u(x) + \omega^2 \rho(x) u(x) = 0, & x \in \Omega \ \partial_{\nu} u(x) = g(x), & x \in \partial\Omega \end{cases}$$9

as $$\mathcal{L}_{\lambda,\mu} u = \mu \Delta u + (\lambda+\mu) \nabla(\nabla\cdot u), \quad \partial_\nu u = [\lambda (\nabla\cdot u) I + 2\mu \nabla^s u]\nu,$$0. The key formula relating measurements to the Fourier transform of $$\mathcal{L}_{\lambda,\mu} u = \mu \Delta u + (\lambda+\mu) \nabla(\nabla\cdot u), \quad \partial_\nu u = [\lambda (\nabla\cdot u) I + 2\mu \nabla^s u]\nu,$$1 is

$$\mathcal{L}_{\lambda,\mu} u = \mu \Delta u + (\lambda+\mu) \nabla(\nabla\cdot u), \quad \partial_\nu u = [\lambda (\nabla\cdot u) I + 2\mu \nabla^s u]\nu,$$2

Consequently, the boundary measurement inner products for different frequencies $$\mathcal{L}_{\lambda,\mu} u = \mu \Delta u + (\lambda+\mu) \nabla(\nabla\cdot u), \quad \partial_\nu u = [\lambda (\nabla\cdot u) I + 2\mu \nabla^s u]\nu,$$3 furnish explicit samples of $$\mathcal{L}_{\lambda,\mu} u = \mu \Delta u + (\lambda+\mu) \nabla(\nabla\cdot u), \quad \partial_\nu u = [\lambda (\nabla\cdot u) I + 2\mu \nabla^s u]\nu,$$4: $$\mathcal{L}_{\lambda,\mu} u = \mu \Delta u + (\lambda+\mu) \nabla(\nabla\cdot u), \quad \partial_\nu u = [\lambda (\nabla\cdot u) I + 2\mu \nabla^s u]\nu,$$5 The inverse Fourier integral recovers the spatial density: $$\mathcal{L}_{\lambda,\mu} u = \mu \Delta u + (\lambda+\mu) \nabla(\nabla\cdot u), \quad \partial_\nu u = [\lambda (\nabla\cdot u) I + 2\mu \nabla^s u]\nu,$$6

5. Constructive Density-Reconstruction Scheme

The global density-reconstruction algorithm proceeds as follows:

Step	Description	Output/Operation
1	Inject $$\mathcal{L}_{\lambda,\mu} u = \mu \Delta u + (\lambda+\mu) \nabla(\nabla\cdot u), \quad \partial_\nu u = [\lambda (\nabla\cdot u) I + 2\mu \nabla^s u]\nu,$$7 high-density inclusions of radius $$\mathcal{L}_{\lambda,\mu} u = \mu \Delta u + (\lambda+\mu) \nabla(\nabla\cdot u), \quad \partial_\nu u = [\lambda (\nabla\cdot u) I + 2\mu \nabla^s u]\nu,$$8 in a subregion of $$\mathcal{L}_{\lambda,\mu} u = \mu \Delta u + (\lambda+\mu) \nabla(\nabla\cdot u), \quad \partial_\nu u = [\lambda (\nabla\cdot u) I + 2\mu \nabla^s u]\nu,$$9. Measure $\lambda, \mu > 0$0 (reference) and $\lambda, \mu > 0$1 (perturbed) at resonance frequency $\lambda, \mu > 0$2.	Boundary N–D maps acquired
2	Compute (or approximate) the homogenized map $\lambda, \mu > 0$3. Check $\lambda, \mu > 0$4 as $\lambda, \mu > 0$5.	Effective N–D map validated
3	For each traction $\lambda, \mu > 0$6, solve $\lambda, \mu > 0$7 in $\lambda, \mu > 0$8, $\lambda, \mu > 0$9; evaluate $\rho(x) > 0$0.	Right-hand side vector formed
4	For a grid of frequencies $\rho(x) > 0$1, select CGO pairs $\rho(x) > 0$2 yielding fields $\rho(x) > 0$3 as above. Calculate $\rho(x) > 0$4.	Fourier samples $\rho(x) > 0$5
5	Recover $\rho(x) > 0$6 by inverse Fourier sum or other numerical inversion of $\rho(x) > 0$7.	Reconstructed density $\rho(x) > 0$8

This constructive paradigm leverages resonant microstructure to induce analytic tractability in the elastic Calderón inverse problem, enabling direct application of CGO-based Fourier reconstruction techniques.

6. Conceptual Significance and Analytic Implications

By embedding a periodic array of resonant high-density inclusions and tuning frequency to the resonance of an inclusion-specific Newton–Kelvin eigenvalue, one attains a homogenized elastic system with a uniform negative density shift. This background effect linearizes the typically nonlinear dependency of the N–D map with respect to $\rho(x) > 0$9, making global density reconstruction feasible through analytic means. This scheme provides a metamaterial-inspired analytic framework for elastic coefficient inverse problems and delineates a concrete route for leveraging nanoscale resonators in reconstruction algorithms (Diao et al., 16 Jan 2026).

A plausible implication is that similar principles could extend to other classes of coefficient inverse problems in PDEs, where homogenized metamaterial effects facilitate linearization and render complex inverse procedures tractable within an analytic paradigm.

7. Relation to Broader Research Areas

The elastic Calderón-type inverse problem is a direct analogue to the electrical Calderón problem (or Electrical Impedance Tomography) in the context of linear elasticity. The approach via hard inclusions and negative-density shifts draws from advances in metamaterials and resonant microstructure engineering. The analytic use of CGO solutions for Fourier sampling aligns this methodology with established uniqueness and reconstruction frameworks in inverse problems. The results by Diao, Sini, and Tang set a precedent for constructive, physically-inspired approaches to elasticity inverse coefficient problems, bridging advances in analytical techniques with transformative materials-based strategies (Diao et al., 16 Jan 2026).