Inverse Density Problem for Linear Elasticity: Uniqueness from Local Measurements on a Partially Accessible Boundary

Abstract: We consider the inverse boundary value problem in an elasticity system. It is proved that the density function $ρ$ and its derivatives at the boundary can be uniquely determined from the local Cauchy data. Furthermore, if the density function is analytic, we can uniquely determine the internal buried objects, as well as the unknown boundary and the boundary conditions imposed on it. Our methods mainly based on a precise characterization for the principal part of the difference between a special first-order singular solution and the fundamental solution in the $H^m$ norm, and the blow-up property for the boundary Sobolev norms of the volume potential corresponding to the fundamental solution.

• The paper establishes that all traces and derivatives of the mass density on accessible boundaries are uniquely determined from local Cauchy data.
• The authors introduce refined singularity analysis and Sobolev norm blow-up techniques to overcome weak singularity challenges inherent in elasticity systems.
• Under an analyticity assumption, the study uniquely recovers internal inclusions, unknown boundaries, and boundary conditions, offering insights for nondestructive testing and seismic imaging.

Uniqueness in the Inverse Density Problem for Linear Elasticity with Partial Boundary Data

Problem Setting and Main Results

The paper addresses the inverse boundary value problem (IBVP) for the time-harmonic linear elasticity system in three dimensions. Specifically, it investigates uniqueness in determining the mass density function $\rho$ (including all its boundary derivatives), internal inclusions, unknown boundaries, and associated boundary conditions, using only local Cauchy data on a subset $\Gamma$ of a partially accessible boundary.

Let $\Omega\subset\mathbb{R}^3$ be a bounded domain with smooth boundary, partitioned into two open, disjoint subsets $\Sigma^{(\Theta)}$ and $\Sigma^{(R)}$. The domain may contain an unknown inclusion $D$. The elastic displacement $\bm{u}$ satisfies the Lamé system with Robin or Dirichlet (possibly mixed) boundary conditions, depending on the sub-boundary. The data provided consists of the Cauchy set $\mathcal{C}^{(\beta_1,\beta_2)}_\Gamma$ associated with the Dirichlet-to-Neumann map, but only on subset $\Gamma \subset\Sigma^{(\Theta)}$.

The main theorem asserts:

• All traces and derivatives of $\rho$ on $\Gamma$0 are uniquely determined by $\Gamma$1.
• If $\Gamma$2 is analytic, then the internal inclusion, the boundary itself, and imposed boundary conditions are also uniquely determined.

The result settles boundary and interior uniqueness for the density and underlying geometry, using highly restricted, local measurements.

Technical Approach

Two main innovations enable the uniqueness result:

1. Precise singularity analysis: The authors construct special first-order singular solutions and analyze the leading asymptotic difference between these solutions and the elastic fundamental solution $\Gamma$3 in local $\Gamma$4 Sobolev norms.
2. Blow-up of boundary Sobolev norms: They characterize the divergence (as source points approach the boundary) of the boundary Sobolev norms of volume potentials for the fundamental solution modulated by the mass density. Crucially, they resolve the technical challenge that the elastic double-layer kernel is only weakly singular for non-physical values of parameter pairs $\Gamma$5, using a stress vector identity.

The analytical techniques rely on fine properties of layer potentials and singular solutions for elasticity, following strategies reminiscent of previous work on the Calderón problem and its vectorial analogues, but significantly refined to handle the added geometric and regularity complexity.

Core Theorems and Methodological Details

Boundary Determination of Density

Let $\Gamma$6 and consider a sequence of singular sources $\Gamma$7 from within $\Gamma$8. The main technical estimate precisely quantifies the difference between the solution of the boundary value problem with singular boundary data (localized near $\Gamma$9) and the fundamental solution, in both bulk and boundary Sobolev norms. This iterative scheme allows for recursive control of higher derivatives.

Furthermore, for any function $\Omega\subset\mathbb{R}^3$0, the authors show that if all tangential derivatives up to order $\Omega\subset\mathbb{R}^3$1 vanish but the $\Omega\subset\mathbb{R}^3$2-th order normal derivative does not, then the associated boundary Sobolev norms for the volume potential (with kernel involving the fundamental solution) exhibit blow-up as the source approaches the boundary. This property is then used for contradiction arguments in the uniqueness proofs.

Propagation of Uniqueness and Domain Recovery

The proof proceeds in three main claims:

• Equivalence of Cauchy data and stress vectors: For physical values of the Lamé parameters and Robin/Dirichlet data, the generalized stress vector on the boundary can be related to Cauchy data for singular solutions.
• Uniqueness of all boundary derivatives of $\Omega\subset\mathbb{R}^3$3: Through recursive application of blow-up results and regularity of solutions, all order normal derivatives at an arbitrary boundary point are uniquely determined.
• Recovery of analytic $\Omega\subset\mathbb{R}^3$4, inclusions, and unknown boundaries: Analyticity enables unique continuation into the domain. By constructing appropriate Green's functions and using unique continuation, the geometry (inclusion $\Omega\subset\mathbb{R}^3$5 and sub-boundaries) and boundary conditions—up to the imposed local data—are determined uniquely.

The argument follows a layered structure, first excluding the possibility of distinct boundary jets for the density, then propagating uniqueness via analyticity and unique continuation to the whole accessible domain, and finally using boundary layer singularity to exclude multiple geometric configurations.

Relation to Prior Literature

The work fits within the classical program initiated by Calderón [APC1980] and its elasticity analogues [GNGU1994, Nakamura1993]. Uniqueness for the Lamé moduli from full or partial Cauchy data, and for density under various regularity or perturbative assumptions, has been an area of sustained interest [BFPRV2018, Imanuvilov2012, Eskin2002]. However, previous approaches (CGO, singular solutions, microlocal analysis) encountered obstacles when faced with partially accessible boundaries, unknown inclusions, or analytic regularity. The paper’s methodology particularly extends the singular solution framework, addressing challenges associated with the elasticity system's operator structure and real-world boundary configurations.

Implications and Perspectives

This result has significant theoretical implications for the identifiability of material properties and domain structure using limited boundary data in elasticity. Inverse problems under partial data and unknown geometry are central to practical applications such as nondestructive evaluation and seismic imaging, where data coverage is realistically always incomplete. The analytic regime assumption, though restrictive, is natural for many engineering materials.

On a methodological level, the characterization of the boundary behavior of singular volume potentials, and the associated blow-up analysis in Sobolev spaces, may influence future developments in both elasticity and broader classes of PDE-constrained inverse problems. Specifically, the connection between high-order boundary jets and the singularity structure of volume potentials opens new avenues for non-perturbative uniqueness proofs.

For computational methods, the precise boundary estimates can inform regularization strategies and sensitivity analysis for numerical inversion procedures, especially in the presence of geometric uncertainty or limited-access measurements.

Conclusion

The paper establishes boundary and interior uniqueness for the inverse density problem in the linear elasticity system from local measurements, even in the presence of unknown boundaries and internal inclusions. The key tools are a refined singular solution approach in $\Omega\subset\mathbb{R}^3$6 norms and a boundary blow-up argument for volume potentials, extending the boundary determination paradigm to a highly general, physically motivated context. Under an analyticity assumption on the density, not only is the density itself but also the precise geometry and boundary conditions of the domain are uniquely determined by local Cauchy data.

Citation: "Inverse Density Problem for Linear Elasticity: Uniqueness from Local Measurements on a Partially Accessible Boundary" (2607.00639)