We study an elastic version of the Calderón problem: determine the internal mass density <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$\rho (\textbf{x})$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>ρ</mi> <mo stretchy="false">(</mo> <mi mathvariant="bold">x</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> from the Neumann-to-Dirichlet (N-D) map associated with the isotropic Lamé system <Equation ID="Equ338"> <EquationSource Format="TEX">$ \mathcal {L}_{\lambda ,\mu } \textbf{u} + \omega ^2 \rho (\textbf{x}) \textbf{u} = \textbf{0} $</EquationSource> <EquationSource Format="MATHML"><math display="block"> <mrow> <msub> <mi mathvariant="script">L</mi> <mrow> <mi>λ</mi> <mo>,</mo> <mi>μ</mi> </mrow> </msub> <mi mathvariant="bold">u</mi> <mo>+</mo> <msup> <mi>ω</mi> <mn>2</mn> </msup> <mi>ρ</mi> <mrow> <mo stretchy="false">(</mo> <mi mathvariant="bold">x</mi> <mo stretchy="false">)</mo> </mrow> <mi mathvariant="bold">u</mi> <mo>=</mo> <mn mathvariant="bold">0</mn> </mrow> </math></EquationSource> </Equation>in a bounded elastic body <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$\Omega \subset \mathbb {R}^3$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi mathvariant="normal">Ω</mi> <mo>⊂</mo> <msup> <mrow> <mi mathvariant="double-struck">R</mi> </mrow> <mn>3</mn> </msup> </mrow> </math></EquationSource> </InlineEquation>. To the best of our knowledge, this work provides the first constructive strategy, based on embedding resonant hard inclusions, for the Calderón-type inverse problem in the isotropic Lamé system to reconstruct the density <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$\rho $</EquationSource> <EquationSource Format="MATHML"><math> <mi>ρ</mi> </math></EquationSource> </InlineEquation>. The key to our strategy is to induce a uniform negative shift in the effective density (i.e., a negative effective density) by embedding a subwavelength periodic array of resonant high-density inclusions. We insert a periodic cluster of high-density inclusions of size a and density <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$\rho _1 \simeq a^{-2}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>ρ</mi> <mn>1</mn> </msub> <mo>≃</mo> <msup> <mi>a</mi> <mrow> <mo>-</mo> <mn>2</mn> </mrow> </msup> </mrow> </math></EquationSource> </InlineEquation> into <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$\Omega $</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="normal">Ω</mi> </math></EquationSource> </InlineEquation>, away from <InlineEquation ID="IEq6"> <EquationSource Format="TEX">$\partial \Omega $</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>∂</mi> <mi mathvariant="normal">Ω</mi> </mrow> </math></EquationSource> </InlineEquation>. For excitation frequencies <InlineEquation ID="IEq7"> <EquationSource Format="TEX">$\omega $</EquationSource> <EquationSource Format="MATHML"><math> <mi>ω</mi> </math></EquationSource> </InlineEquation> tuned to a suitable eigenvalue of the elastic Newton operator (i.e., Kelvin operator) associated with a single inclusion, we show that the N-D map <InlineEquation ID="IEq8"> <EquationSource Format="TEX">$\Lambda _D$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi mathvariant="normal">Λ</mi> <mi>D</mi> </msub> </math></EquationSource> </InlineEquation> of the composite medium converges, as <InlineEquation ID="IEq9"> <EquationSource Format="TEX">$a \rightarrow 0$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>a</mi> <mo stretchy="false">→</mo> <mn>0</mn> </mrow> </math></EquationSource> </InlineEquation> and the number M of inclusions tends to infinity, to an effective map <InlineEquation ID="IEq10"> <EquationSource Format="TEX">$\Lambda _{\mathcal {P}}$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi mathvariant="normal">Λ</mi> <mi mathvariant="script">P</mi> </msub> </math></EquationSource> </InlineEquation> corresponding to an elastic medium with a uniform negative density shift <InlineEquation ID="IEq11"> <EquationSource Format="TEX">$-\mathcal {P}^2$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo>-</mo> <msup> <mrow> <mi mathvariant="script">P</mi> </mrow> <mn>2</mn> </msup> </mrow> </math></EquationSource> </InlineEquation>. We prove an operator norm estimate <Equation ID="Equ339"> <EquationSource Format="TEX">$ \Vert \Lambda _D - \Lambda _{\mathcal {P}}\Vert \le C a^{\alpha } \mathcal {P}^6, $</EquationSource> <EquationSource Format="MATHML"><math display="block"> <mrow> <mrow> <mo stretchy="false">‖</mo> </mrow> <msub> <mi mathvariant="normal">Λ</mi> <mi>D</mi> </msub> <mo>-</mo> <msub> <mi mathvariant="normal">Λ</mi> <mi mathvariant="script">P</mi> </msub> <mrow> <mo stretchy="false">‖</mo> <mo>≤</mo> <mi>C</mi> </mrow> <msup> <mi>a</mi> <mi>α</mi> </msup> <msup> <mrow> <mi mathvariant="script">P</mi> </mrow> <mn>6</mn> </msup> <mo>,</mo> </mrow> </math></EquationSource> </Equation>with <InlineEquation ID="IEq12"> <EquationSource Format="TEX">$\alpha > 0$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>α</mi> <mo>></mo> <mn>0</mn> </mrow> </math></EquationSource> </InlineEquation> depending on the geometric scaling. We then derive a first-order linearization formula for <InlineEquation ID="IEq13"> <EquationSource Format="TEX">$\Lambda _{\mathcal {P}}$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi mathvariant="normal">Λ</mi> <mi mathvariant="script">P</mi> </msub> </math></EquationSource> </InlineEquation> around this negative background, expressed in terms of <InlineEquation ID="IEq14"> <EquationSource Format="TEX">$\rho $</EquationSource> <EquationSource Format="MATHML"><math> <mi>ρ</mi> </math></EquationSource> </InlineEquation> and the Newton volume potential for the shifted Lamé operator. By testing this linearized relation with suitable complex geometric optics solutions for the Lamé system, we obtain a reconstruction formula for the Fourier transform of <InlineEquation ID="IEq15"> <EquationSource Format="TEX">$\rho $</EquationSource> <EquationSource Format="MATHML"><math> <mi>ρ</mi> </math></EquationSource> </InlineEquation>, and hence a global density recovery scheme. The method proposed in this paper demonstrates how metamaterial-inspired effective media can be exploited as an analytic tool for inverse coefficient problems in linear elasticity, enabling a tractable linearization around a negative background and an explicit global reconstruction procedure. This provides a novel strategy and paradigm for using nanoscale metamaterials to solve inverse problems.

Elastic Calderón problem via resonant hard inclusions: linearisation of the Neumann-to-Dirichlet map and density reconstruction

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发表日期：

2026年5月16日

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Huaian Diao,
Mourad Sini,
Ruixiang Tang

摘要

We study an elastic version of the Calderón problem: determine the internal mass density $\rho (\textbf{x})$ $ρ (x)$ from the Neumann-to-Dirichlet (N-D) map associated with the isotropic Lamé system $ \mathcal {L}_{\lambda ,\mu } \textbf{u} + \omega ^2 \rho (\textbf{x}) \textbf{u} = \textbf{0} $ $L_{λ, μ} u + ω^{2} ρ (x) u = 0$ in a bounded elastic body $\Omega \subset \mathbb {R}^3$ $Ω \subset R^{3}$ . To the best of our knowledge, this work provides the first constructive strategy, based on embedding resonant hard inclusions, for the Calderón-type inverse problem in the isotropic Lamé system to reconstruct the density $\rho $ $ρ$ . The key to our strategy is to induce a uniform negative shift in the effective density (i.e., a negative effective density) by embedding a subwavelength periodic array of resonant high-density inclusions. We insert a periodic cluster of high-density inclusions of size a and density $\rho _1 \simeq a^{-2}$ $ρ_{1} ≃ a^{- 2}$ into $\Omega $ $Ω$ , away from $\partial \Omega $ $\partial Ω$ . For excitation frequencies $\omega $ $ω$ tuned to a suitable eigenvalue of the elastic Newton operator (i.e., Kelvin operator) associated with a single inclusion, we show that the N-D map $\Lambda _D$ $Λ_{D}$ of the composite medium converges, as $a \rightarrow 0$ $a \to 0$ and the number M of inclusions tends to infinity, to an effective map $\Lambda _{\mathcal {P}}$ $Λ_{P}$ corresponding to an elastic medium with a uniform negative density shift $-\mathcal {P}^2$ $- P^{2}$ . We prove an operator norm estimate $ \Vert \Lambda _D - \Lambda _{\mathcal {P}}\Vert \le C a^{\alpha } \mathcal {P}^6, $ $‖ Λ_{D} - Λ_{P} ‖ \leq C a^{α} P^{6},$ with $\alpha > 0$ $α > 0$ depending on the geometric scaling. We then derive a first-order linearization formula for $\Lambda _{\mathcal {P}}$ $Λ_{P}$ around this negative background, expressed in terms of $\rho $ $ρ$ and the Newton volume potential for the shifted Lamé operator. By testing this linearized relation with suitable complex geometric optics solutions for the Lamé system, we obtain a reconstruction formula for the Fourier transform of $\rho $ $ρ$ , and hence a global density recovery scheme. The method proposed in this paper demonstrates how metamaterial-inspired effective media can be exploited as an analytic tool for inverse coefficient problems in linear elasticity, enabling a tractable linearization around a negative background and an explicit global reconstruction procedure. This provides a novel strategy and paradigm for using nanoscale metamaterials to solve inverse problems.