Approximation and Carleman Formulas for Solutions to Parabolic Lamé-Type Operators in Cylindrical Domains

Abstract

Assume that $s\in{𝕅}$ and $T_{1},T_{2}\in{𝕉}$, with $T_{1}<T_{2}$. Assume further that $\Omega$ and $\omega$ are bounded domains in ${𝕉}^{n}$, 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 $\omega\times(T_{1},T_{2})\subset{𝕉}^{n+1}$ 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.