Lipschitz stability in the lateral Cauchy problem for elasticity system
The authors deal with the problem of determining the displacement vector \({\mathbf u}\) in the isotropic elasticity system \[ \rho D_t^2{\mathbf u}-\mu[\Delta {\mathbf u}+\nabla(\nabla^T {\mathbf u})]-\nabla(\lambda \nabla^T{\mathbf u}) -\sum_{j=1}^3\,\nabla \mu\cdot (\nabla u_j+D_{x_j}{\mathbf u}){\mathbf e}_j,\quad \text{ in\;} \Omega\times (0,T)=: Q, \tag{1} \] subject to the lateral Cauchy conditions \[ {\mathbf u}={\mathbf g} \quad \text{ on\;} \partial \Omega\times (0,T),\qquad D_\nu{\mathbf u}={\mathbf h} \quad \text{ on\;} \Gamma\times (0,T). \tag{2} \] They assume that \(\Omega\) is a bounded domain with a \(C^2\)-boundary and \(\Gamma\) is a (smooth) subset of \(\partial \Omega\). Further, they assume that \(\rho\in C^2({\overline Q})\), \(\lambda,\mu\in C^3({\overline Q})\) and \(\rho(x)>0\), \(3\lambda(x)+\mu(x)>0\) for all \(x\in {\overline Q}\). Using a procedure based on Carleman estimates and diagonalization of the principal part of system (1), which makes use of the auxiliary function \(v=\nabla^T{\mathbf u}\), under suitable assumptions on \(\Omega\), \(\Gamma\) and functions \(a=\rho/\mu\) and \(b=\rho/(\lambda+2\mu)\), the authors prove the following estimate \[ \begin{aligned} \| D_t{\mathbf u}(\cdot,t)\| _{H^2(\Omega)} &+ \| {\mathbf u}(\cdot,t)\| _{H^1(\Omega)} + \| D_t\nabla^T{\mathbf u}(\cdot,t)\| _{H^2(\Omega)} + \| \nabla^T{\mathbf u}(\cdot,t)\| _{H^1(\Omega)}\\ \leq C\{ \| {\mathbf h}\| _{H^2(\Gamma \times (0,T))} &+ \| {\mathbf g}\| _{H^1(\partial \Omega\times (0,T))} + \| \nabla^T{\mathbf u}\| _{H^1(\partial \Omega)\times (0,T))} + \| D_\nu\nabla^T{\mathbf u}\| _{H^1(\Gamma \times (0,T))} \}. \end{aligned} \tag{3} \] Moreover, they get rid of the term \(\nabla^T{\mathbf u}\) occurring in the right-hand side in (3), when the lateral conditions (2) are replaced with \[ {\mathbf u}={\mathbf 0} \quad \text{ on\;} \partial \Omega\times (0,T),\qquad \sigma({\mathbf u})\nu={\mathbf h} \quad \text{ on\;} \Gamma\times (0,T), \] where \(\sigma({\mathbf u})=\lambda(\nabla^T)({\mathbf e}_1,{\mathbf e}_2,{\mathbf e}_3) + \mu(\nabla u_1 +D_{x_1}{\mathbf u},\nabla u_2+D_{x_2}{\mathbf u},\nabla u_3+D_{x_3}{\mathbf u})\), provided \({\mathbf u},(\nabla^T)({\mathbf u})\in C^2({\overline Q})^3\).
scientific article; zbMATH DE number 2124048