{
  "id": "global-uniqueness-for-isotropic-elasticity",
  "title": "Global uniqueness for isotropic elasticity",
  "description": "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.",
  "units": [
    {
      "id": "the-displacement-to-traction-map",
      "title": "The displacement-to-traction map",
      "reader": "the-displacement-to-traction-map.html",
      "source": "src/the-displacement-to-traction-map.md",
      "source_sha256": "7f720dc002bd8b742095eda8c91754b0e0b6343f762b8959d4735049885a5de6",
      "status": "author_self_checked"
    },
    {
      "id": "boundary-jets-and-the-transfer-of-solutions",
      "title": "Boundary jets and the transfer of solutions",
      "reader": "boundary-jets-and-the-transfer-of-solutions.html",
      "source": "src/boundary-jets-and-the-transfer-of-solutions.md",
      "source_sha256": "f3699d045af336742c45365a040fe9285f407b106eea25d7590b74a1a5f37adb",
      "status": "author_self_checked"
    },
    {
      "id": "an-augmented-system-and-a-carleman-estimate",
      "title": "An augmented system and a Carleman estimate",
      "reader": "an-augmented-system-and-a-carleman-estimate.html",
      "source": "src/an-augmented-system-and-a-carleman-estimate.md",
      "source_sha256": "dbdba6f72bc19462888584b63729a4ee6fb9b3fd8159285e9cab376915e997a3",
      "status": "author_self_checked"
    },
    {
      "id": "planar-transport-equations",
      "title": "Planar transport equations",
      "reader": "planar-transport-equations.html",
      "source": "src/planar-transport-equations.md",
      "source_sha256": "6e52b638ad3de7a9a20d63947e565d9b234c2d63dd9c9b10c189439f9642e198",
      "status": "author_self_checked"
    },
    {
      "id": "exact-solutions-with-prescribed-leading-terms",
      "title": "Exact solutions with prescribed leading terms",
      "reader": "exact-solutions-with-prescribed-leading-terms.html",
      "source": "src/exact-solutions-with-prescribed-leading-terms.md",
      "source_sha256": "8e31618c3094235520342e2dd4fad6410e12bdf49f2303434199729625cd9f8e",
      "status": "author_self_checked"
    },
    {
      "id": "matching-the-transports",
      "title": "Matching the transports",
      "reader": "matching-the-transports.html",
      "source": "src/matching-the-transports.md",
      "source_sha256": "6db0a98cecb194f5f177b23ec1297aaa92b9b5dddc5d4b55acb21ba109d3237e",
      "status": "author_self_checked"
    },
    {
      "id": "the-null-conic-and-the-recovery-of-the-moduli",
      "title": "The null conic and the recovery of the moduli",
      "reader": "the-null-conic-and-the-recovery-of-the-moduli.html",
      "source": "src/the-null-conic-and-the-recovery-of-the-moduli.md",
      "source_sha256": "2319148f0b5bfb9ad25a39957053132c1d85b0534da08dfd2ed96830e5728a37",
      "status": "author_self_checked"
    }
  ],
  "prerequisites": [
    "core course Functional Analysis (Erdman): Hilbert spaces and the Riesz-Frechet representation theorem (Section 4.4, Theorem 4.5.2)",
    "core course Complex Analysis (Lebl): the identity theorem (Theorem 2.4.7), the Cauchy-Pompeiu formula (Theorem 4.6.3), Liouville's theorem (Theorem 3.3.10)",
    "core course Vector Calculus (CLP-4): the divergence theorem (Theorem 4.2.2)",
    "Lebesgue integration, L^2 spaces, distributions, the Fourier transform on R^n and Sobolev spaces H^1, H^1_0 and H^k",
    "programme course Elliptic operators and boundary problems: Fourier transforms and Sobolev norms (lesson Fourier transforms, finite spectra and convex separation, Sections 2 and 7), compactness and smoothness of Sobolev functions (lesson Symbols, finite defects, and the index on a closed manifold, Section 1)",
    "programme course DG-FND: weak subsequences in Hilbert space (lesson Complex manifolds and Kähler metrics, Lemma K.2)"
  ],
  "status": "author_self_checked",
  "proof_providers": "providers.json",
  "provenance": "provenance.json",
  "licence": "CC0-1.0"
}