On approximation of solutions to the heat equation from Lebesgue class $L^2$ by more regular solutions

Abstract

Let $s \in {\mathbb N}$, $T_1,T_2 \in {\mathbb R}$, $T_1<T_2$, and $Ω, ω$ be bounded domains in ${\mathbb R}^n$, $n \geq 1$, such that $ω\subset Ω$ and the complement $Ω\setminus ω$ has no (non-empty) compact components in $Ω$. We prove that this is the necessary and sufficient condition for the space $H^{2s,s} _{\mathcal H} (Ω\times (T_1,T_2))$ of solutions to the heat operator ${\mathcal H} $ in a cylinder domain $Ω\times (T_1,T_2)$ from the anisotropic Sobolev space $H^{2s,s} (Ω\times (T_1,T_2))$ to be dense in the space $L^{2} _{\mathcal H}(ω\times (T_1,T_2))$, consisting of solutions in the domain $ω\times (T_1,T_2)$ from the Lebesgue class $L^{2} (ω\times (T_1,T_2))$. As an important corollary we obtain the theorem on the existence of a basis with the double orthogonality property for the pair of the Hilbert spaces $H^{2s,s} _{\mathcal H} (Ω\times (T_1,T_2))$ and $L^{2} _{\mathcal H}(ω\times (T_1,T_2))$ .