Approximation of solutions to parabolic Lam\'e type operators in cylinder domains and Carleman's formulas for them

Abstract

Let $s \in {\mathbb N}$, $T_1,T_2 \in {\mathbb R}$, $T_1<T_2$, and let $\Omega, \omega $ be bounded domains in ${\mathbb R}^n$, $n \geq 1$ such that $\omega \subset \Omega$ and the complement $\Omega \setminus \omega$ have no non-empty compact components in $\Omega$. We investigate the problem of approximation of solutions to parabolic Lam\'e type system from the Lebesgue class $L^2(\omega \times (T_1,T_2))$ in a cylinder domain $\omega \times (T_1,T_2) \subset {\mathbb R}^{n+1}$ by more regular solutions in a bigger domain $\Omega \times (T_1,T_2)$. As an application of the obtained approximation theorems we construct Carleman's formulas for recovering solutions to these parabolic operators from the Sobolev class $H^{2s,s}(\Omega \times (T_1,T_2))$ via values the solutions on a part of the lateral surface of the cylinder and the corresponding them stress tensors.