Lipschitz stability in the lateral Cauchy problem for elasticity system

Abstract

We consider the isotropic elasticity system: \[ \begin{array}{cclcc} \rho \partial _{t}^{2}\mathbf{u}&-&\mu (\Delta\mathbf{u}+\nabla (\nabla ^{T}\mathbf{u})-\nabla (\lambda\nabla ^{T}\mathbf{u})&&\\ &-&\sum _{j=1}^{3}\nabla\mu\cdot (\nabla u_{j}+\partial _{j}\mathbf{u})\mathbf{e}_{j}=0 &\text{in}& \Omega\times (0,T) \end{array} \] for the displacement vector $\mathbf{u} = (u_{1}, u_{2}, u_{3})$ depending on $x \in \Omega$ and $t \in (0, T)$ where $\Omega$ is a bounded domain in $\mathbb{R}^{3}$ with the $C^{2}$-boundary, and we assume the density $\rho \in C^{2}(\Bar{\Omega}\times[0, T])$ and the Lamé parameters $\mu , \lambda \in C^{3}(\Bar{\Omega}\times[0, T])$. We will give Lipschitz stability estimates for solutions $\mathbf{u}$ to the above elasticity system with the lateral boundary data \[ \begin{array}{cc} \mathbf{u} = \mathbf{g} \textrm{ on } \partial\Omega\times (0, T),& \partial _{\nu}\mathbf{u} = \mathbf{h} \textrm{ on } \Gamma \times (0, T) \end{array} \] where $\Gamma$ is some part of $\partial\Omega$. Our proof is based on (1) a Carleman estimate with boundary data, (2) cut-off technique, and (3) principal diagonalization of the Lamé system.