Lipschitz stability in the lateral Cauchy problem for elasticity system

Abstract

We consider the isotropic elasticity system: ρ∂2 t u − μ(Δu + ∇(∇Tu))−∇(λ∇Tu) − 3 X j=1 ∇μ · (∇uj + ∂ju)ej = 0 in Ω× (0, T) for the displacement vector u = (u1, u2, u3) depending on x ∈ Ω and t ∈ (0, T) where Ω is a bounded domain in R3 with the C2-boundary, and we assume the density ρ ∈ C2(Ω× [0, T]) and the Lam´e parameters μ, λ ∈ C3(Ω × [0, T]). We will give Lipschitz stability estimates for solutions u to the above elasticity system with the lateral boundary data u = g on ∂Ω × (0, T), ∂νu = h on Γ × (0, T) where Γ is some part of ∂Ω. Our proof is based on (1) a Carleman estimate with boundary data, (2) cut-off technique, and (3) principal diagonalization of the Lam´e system.