Unique Determination of the Lamé Parameters in Three Dimensions

Abstract

We prove that the full Dirichlet-to-Neumann map for static isotropic elasticity uniquely determines both smooth Lamé parameters in a bounded smooth domain of \(\mathbb R^3\) under strong convexity. The proof uses an exact four-component Laplace-type reduction, but its decisive identities are obtained from physical elastic solutions. The characteristic quotient of the reduced system has rank three, whereas the leading physical adjoint lift supplies only two directions. We construct the missing polarization from the first nonvanishing correction of a reduced amplitude in the kernel of that lift. Paired with fixed primal and adjoint quotient frames, the resulting complex geometrical optics solutions yield a complete family of holomorphically weighted \(3\times3\) matrix moments through the physical Betti identity. Cauchy--Plemelj factorization converts these moments into an invertible relative transport normalized in the common exterior. Holomorphic dependence on the projective null direction and transport rigidity then determine a material ratio and the shear modulus, hence both Lamé parameters. No smallness, symmetry, separability, or analyticity hypothesis is imposed.