On approximation theorems for solutions to strongly parabolic systems in anisotropic Sobolev spaces
Alexander Shlapunov, Pavel Vilkov
TL;DR
The paper tackles Runge-type approximation for solutions to strongly parabolic systems with constant coefficients in anisotropic Sobolev spaces on non-cylindrical domains. It extends Runge-type density results from Lebesgue spaces to the anisotropic Sobolev scale $\mathbf{H}^{\gamma,2ms,s}_{k,\mathcal{L}}(G)$ by proving a necessary-and-sufficient condition: $S_{\mathcal{L}}(\overline{G_2})$ is dense in $\mathbf{H}^{\gamma,2ms,s}_{k,\mathcal{L}}(G_1)$ precisely when the time-slice difference $G_2(t)\setminus G_1(t)$ has no compact components for all $t$. The proof combines Green's formulas, the fundamental solution $\Phi$, and hypoellipticity with duality arguments (Hahn–Banach) to construct a dual distribution and a parabolic potential, ensuring density. This work generalizes Runge-type approximation to anisotropic Sobolev spaces in non-cylindrical domains and connects with prior $L^2$ and Lebesgue-space Runge results, enabling stable boundary-value problem approximations in more refined function spaces.
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.