On Runge type theorems for solutions to strongly uniformly parabolic operators

Abstract

Let $G_1, G_2 $ be domains in ${\mathbb R}^{n+1}$, $n \geq 2$, such that $G_1 \subset G_2$ and the domain $G_1$ have rather regular boundary. We investigate the problem of approximation of solutions to strongly uniformly $2m$-parabolic system $\mathcal L$ in the domain $G_1$ by solutions to the same system in the domain $G_2$. First, we prove that the space $S _{\mathcal L}(G_2)$ of solutions to the system $\mathcal L$ in the domain $G_2$ is dense in the space $S _{\mathcal L}(G_1)$, endowed with the standard Fr\'echet topology of the uniform convergence on compact subsets in $G_1$, if and only if the complements $G_2 (t) \setminus G_1 (t)$ have no non-empty compact components in $G_2 (t)$ for each $t\in \mathbb R$, where $G_j (t) = \{x \in {\mathbb R}^n: (x,t) \in G_j\}$. Next, under additional assumptions on the regularity of the bounded domains $G_1$ and $G_1(t)$, we prove that solutions from the Lebesgue class $L^2(G_1)\cap S _{\mathcal L}(G_1)$ can be approximated by solutions from $S _{\mathcal L}(G_2)$ if and only if the same assumption on the complements $G_2 (t) \setminus G_1 (t)$, $t\in \mathbb R$, is fulfilled.