Global uniqueness for isotropic elasticity
The displacement-to-traction map of an isotropic elastic body determines both Lamé moduli: on a bounded connected domain in R^3 with smooth boundary, smooth moduli with μ > 0 and 3λ + 2μ > 0 are recovered from the map from boundary displacements to boundary tractions. Adjoining the divergence as an unknown gives a system with principal part the Laplacian; exact solutions of complex geometrical optics type have leading terms governed by a transport equation; a comparison of the transport equations of the two media depends holomorphically on the direction, which ranges over a conic in the complex projective plane, and a bundle over this conic forces it to be the identity. The course is an exposition of OpenAI's proof (2026). Boundary determination, interior elliptic regularity, a Carleman estimate for the Laplacian and holomorphic frames over a disk are used as named results from the literature.
Seven lessons with proofs, examples and exercises with solutions. Written and self-checked by Claude Opus 5.5 (Anthropic); independent review is not complete.
- The displacement-to-traction map
- Boundary jets and the transfer of solutions
- An augmented system and a Carleman estimate
- Planar transport equations
- Exact solutions with prescribed leading terms
- Matching the transports
- The null conic and the recovery of the moduli
Theorems and assumptions · Reader and editable sources · Course record
Prerequisites
- Complex manifolds and Kähler metrics (Bundles, curvature and holonomy)
- Symbols, finite defects, and the index on a closed manifold
- Fourier transforms, finite spectra and convex separation
- Lebesgue integration, L2 spaces, distributions, the Fourier transform and the Sobolev spaces H1, H10 and Hk.
- Hilbert spaces and the Riesz representation theorem (core course Functional Analysis); the identity theorem, the Cauchy–Pompeiu formula and Liouville's theorem (core course Complex Analysis).