Real, complex and functional analysis

On Runge type theorems for solutions to strongly uniformly parabolic operators

A. A. Shlapunov^ab, P. Yu. Vilkov<sup>a

a</sup> Siberian Federal University, pr. Svobodnyi, 79, 660041, Krasnoyarsk, Russia
^b Sirius Mathematics Center, Sirius University of Science and Technology, Olimpiyskiy ave. b.1, 354349 Sochi, Russia

Abstract: Let $G_1, G_2 $ be domains with rather regular boundaries in ${\mathbb R}^{n+1}$, $n \geq 2$, such that $G_1 \subset G_2$. 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 uniform convergence on compact subsets in $G_1$, if and only if the sets $G_2 (t) \setminus G_1 (t)$ 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 sets $G_2 (t) \setminus G_1 (t)$, $t\in \mathbb R$, is fulfilled.

Keywords: approximation theorems, Frećhet topologies, strongly uniformly parabolic operators.

UDC: ???.??

MSC: ??X??

Received October 28, 2023, published June 6, 2024

Language: English

DOI: 10.33048/semi.2024.21.029