A Runge theorem for solutions of the heat equation

Abstract:

Let ${\Omega _1}$ and ${\Omega _2}$ be open sets in ${R^n}$ such that ${\Omega _1} \subset {\Omega _2}$. Every solution of the heat equation on ${\Omega _1}$ admits approximation on the compact subsets of ${\Omega _1}$ by functions which satisfy the heat equation throughout ${\Omega _2}$ if and only if this topological condition is met: For every hyperplane $\pi$ in ${R^n}$ orthogonal to the time axis, every compact component of $\pi \backslash {\Omega _1}$ contains a compact component of $\pi \backslash {\Omega _2}$.