On approximation theorems for solutions to strongly parabolic systems in anisotropic Sobolev spaces

TL;DR

This paper studies approximation properties of solutions to strongly parabolic systems in anisotropic Sobolev spaces, focusing on Runge pairs in non-cylindrical domains with constant coefficients and geometric conditions.

Contribution

It establishes criteria for when two domains form a Runge pair for Sobolev solutions of parabolic systems with constant coefficients, based on geometric conditions.

Findings

Runge pair characterization for constant coefficient parabolic systems

Geometric conditions involving domain boundaries and their sections

No compact components in domain complements are necessary and sufficient

Abstract

We investigate the problem on Runge pairs for Sobolev solutions of strongly uniformly parabolic systems in non-cylindrical domains of a special kind. We prove that if the coefficients of a parabolic operator are constant, then two domains with sufficiently smooth boundaries, no parts of which are parallel to the plane $t=0$, form a Runge pair if and only if the complements of any section of the larger domain to the section of the smaller domain by planes $t = const$, have no compact components in the larger section.