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

TL;DR

This paper develops approximation and Carleman formulas for solutions to parabolic Lamé type systems in cylindrical domains, enabling recovery of solutions from boundary data and stress tensors, with implications for inverse problems.

Contribution

It introduces new approximation theorems for parabolic Lamé systems and constructs Carleman formulas for solution recovery from boundary and stress data.

Findings

Approximation of solutions in smaller domains by regular solutions in larger domains.

Construction of Carleman formulas for solution reconstruction.

Application to inverse problems involving stress tensors.

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.