A Runge theorem for solutions of the heat equation

Abstract

Let Ω 1 {\Omega _1} and Ω 2 {\Omega _2} be open sets in R n {R^n} such that Ω 1 ⊂ Ω 2 {\Omega _1} \subset {\Omega _2} . Every solution of the heat equation on Ω 1 {\Omega _1} admits approximation on the compact subsets of Ω 1 {\Omega _1} by functions which satisfy the heat equation throughout Ω 2 {\Omega _2} if and only if this topological condition is met: For every hyperplane π \pi in R n {R^n} orthogonal to the time axis, every compact component of π ∖ Ω 1 \pi \backslash {\Omega _1} contains a compact component of π ∖ Ω 2 \pi \backslash {\Omega _2} .