抛物型Lamé型算子解在柱域上的逼近及其Carleman公式
Approximation of solutions to parabolic Lamé type operators in cylinder domains and Carleman's formulas for them
摘要：
令$s \in {\mathbb N}$、$T_1,T_2 \in {\mathbb R}$、$T_1<T_2$和$Ω, ω$是${\mathbb R}^n$、$n \geq 1$中的有界域，使得$ω\subset Ω$和补$Ω\setminus ω$在$Ω$中没有非空紧分量。研究了Lebesgue类$L^2(ω\times (T_1,T_2))$抛物型Lamé型方程组在圆柱区域$ω\times (T_1,T_2) \subset {\mathbb R}^{n+1}$中的解在更大区域$Ω\times (T_1,T_2)$中的更正则解的逼近问题.作为所得到的逼近定理的应用，我们构造了Carleman的公式，用于从Sobolev类$H^{2s,s}(Ω\times (T_1,T_2))$中恢复这些抛物型算子的解，通过值，在圆柱的一部分侧面上的解和相应的应力张量。
英文摘要：
Let $s \in {\mathbb N}$, $T_1,T_2 \in {\mathbb R}$, $T_1<T_2$, and let $Ω, ω$ be bounded domains in ${\mathbb R}^n$, $n \geq 1$ such that $ω\subset Ω$ and the complement $Ω\setminus ω$ have no non-empty compact components in $Ω$. We investigate the problem of approximation of solutions to parabolic Lamé type system from the Lebesgue class $L^2(ω\times (T_1,T_2))$ in a cylinder domain $ω\times (T_1,T_2) \subset {\mathbb R}^{n+1}$ by more regular solutions in a bigger domain $Ω\times (T_1,T_2)$. As an application of the obtained approximation theorems we construct Carleman's formulas for recovering solutions to these parabolic operators from the Sobolev class $H^{2s,s}(Ω\times (T_1,T_2))$ via values the solutions on a part of the lateral surface of the cylinder and the corresponding them stress tensors.
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