On Runge type theorems for solutions to strongly uniformly parabolic operators

Abstract

Let $G_1,G_2$ be domains in ${\mathbb{R}}^{n+1}$, $n\ge 2$, 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 Fréchet 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.