On approximation of solutions to the heat equation from Lebesgue class $L^2$ by more regular solutions

TL;DR

This paper establishes conditions under which solutions to the heat equation in a larger domain can approximate solutions in a smaller domain within Lebesgue spaces, with implications for basis existence in related Hilbert spaces.

Contribution

It provides a necessary and sufficient condition for the density of regular heat solutions in Lebesgue class solutions on subdomains, extending approximation theory for the heat equation.

Findings

Characterizes when solutions in a larger domain densely approximate solutions in a smaller domain.

Proves the existence of a basis with double orthogonality for specific solution spaces.

Identifies geometric conditions on domains for approximation properties.

Abstract

Let $s \in {\mathbb N}$, $T_1,T_2 \in {\mathbb R}$, $T_1<T_2$, and $\Omega, \omega $ be bounded domains in ${\mathbb R}^n$, $n \geq 1$, such that $\omega \subset \Omega$ and the complement $\Omega \setminus \omega$ has no (non-empty) compact components in $\Omega$. We prove that this is the necessary and sufficient condition for the space $H^{2s,s} _{\mathcal H} (\Omega \times (T_1,T_2))$ of solutions to the heat operator ${\mathcal H} $ in a cylinder domain $\Omega \times (T_1,T_2)$ from the anisotropic Sobolev space $H^{2s,s} (\Omega \times (T_1,T_2))$ to be dense in the space $L^{2} _{\mathcal H}(\omega \times (T_1,T_2))$, consisting of solutions in the domain $\omega \times (T_1,T_2)$ from the Lebesgue class $L^{2} (\omega \times (T_1,T_2))$. As an important corollary we obtain the theorem on the existence of a basis with the double orthogonality property for the pair of the Hilbert spaces $H^{2s,s} _{\mathcal H} (\Omega \times (T_1,T_2))$ and $L^{2} _{\mathcal H}(\omega \times (T_1,T_2))$ .