Uniqueness and Lipschitz Stability of an Inverse Boundary Value Problem for Time-harmonic Elastic Waves

where Ω is an open and bounded domain with smooth boundary, ∇̂u denotes the strain tensor, ∇̂u := 12 (∇u + (∇u)T ), ψ ∈ H(∂Ω) is the boundary displacement or source, and C ∈ L(Ω) denotes the isotropic elasticity tensor with Lamé parameters λ, μ: C = λI3 ⊗ I3 + 2μIsym, a.e. in Ω, where I3 is 3 × 3 identity matrix and Isym is the fourth order tensor such that IsymA = Â, ρ ∈ L(Ω) is the density, and ω is the frequency. Here, we make use of the following notation for matrices and tensors: For 3× 3 matrices A and B we set A : B = ∑3 i,j=1 AijBij and  = 1 2 (A+ A T ). We assume that 0 < α0 ≤ μ ≤ α 0 , 0 < β0 ≤ 2μ+ 3λ ≤ β 0 a.e. in Ω, (1.2) 0 ≤ ρ ≤ γ 0 . (1.3)