OAI.Elasticity.global_uniqueness

The theorem states the defined proposition MainClaim, a global uniqueness statement for isotropic linear elasticity in three dimensions. Let Ω be a subset of ℝ³ that is open, connected, bounded, and has smooth boundary, meaning that each boundary point has a C^∞ diffeomorphism chart, with C^∞ inverse, from a neighborhood onto its image that sends the point to a point whose first coordinate is 0 and under which membership in Ω is exactly positivity of the first coordinate. Let (λ₁, μ₁) and (λ₂, μ₂) be two pairs of Lamé-type coefficient functions, each smooth on some open neighborhood of the closure of Ω and satisfying μ > 0 and 3λ + 2μ > 0 at every point of the closure. Here H¹(Ω) is the closure, in L²(Ω) paired with three L² derivative components, of the jets (f, ∂f) of C^∞ vector fields f with square-integrable f and derivatives. Traces are the classes of H¹ elements modulo the closure of jets of smooth compactly supported fields with support in Ω. The energy pairing of u and v is the integral over Ω of λ(div u)(div v) + 2μ times the sum over i,j of the strain products, where the strain is the symmetrized derivative. The Dirichlet-to-Neumann pairing DN(f,g) is the energy of a chosen weak solution with trace f (weak meaning energy zero against all zero-trace test elements, with an arbitrary representative of f used if none exists) against a chosen representative of g. If DN for (λ₁, μ₁) equals DN for (λ₂, μ₂), then λ₁ = λ₂ and μ₁ = μ₂ at every point of Ω. The quotient norm on boundary data is not identified with fractional Sobolev norms.