Approximation and Carleman formulas for solutions to parabolic Lamé-type operators in cylindrical domains

Assume that $s \in {\Bbb N}$ and $T_1,T_2 \in {\Bbb R}$, with $T_1<T_2$. Assume further that $\Omega$ and $\omega $ are bounded domains in ${\Bbb R}^n$, with $n \geq 1$, such that $\omega \subset \Omega$ and the complement $\Omega \setminus \omega$ has no nonempty compact components in $\Omega$. We study the approximation of solutions in the Lebesgue space $L^2(\omega \times (T_1,T_2))$ to parabolic Lamé-type operators in the cylindrical domain $\omega \times (T_1,T_2) \subset {\Bbb R}^{n+1}$ by more regular solutions in the larger domain $\Omega \times (T_1,T_2)$. As application of the approximation theorems, we construct some Carleman formulas for recovering solutions to these parabolic operators in the Sobolev space $H^{2s,s}(\Omega \times (T_1,T_2))$ via the values of the solutions and the corresponding stress tensors on a part of the lateral surface of the cylinder.