Analytic properties of heat equation solutions and reachable sets

Abstract

There recently has been some interest in the space of functions on an interval satisfying the heat equation for positive time in the interior of this interval. Such functions were characterised as being analytic on a square with the original interval as its diagonal. In this short note we provide a direct argument that the analogue of this result holds in any dimension. For the heat equation on a bounded Lipschitz domain $\Omega \subset \mathbb{R}^d$ at positive time all solutions are analytically extendable to a geometrically determined subdomain $\mathcal{E}(\Omega)$ of $\mathbb{C}^d$ containing $\Omega$. This domain is sharp in the sense that there is no larger domain for which this is true. If $\Omega$ is a ball we prove an almost converse of this theorem. Any function that is analytic in an open neighborhood of $\mathcal{E}(\Omega)$ is reachable in the sense that it can be obtained from a solution of the heat equation at positive time. This is based on an analysis of the convergence of heat equation solutions in the complex domain using the boundary layer potential method for the heat equation. The converse theorem is obtained using a Wick rotation into the complex domain that is justified by our results. This gives a simple explanation for the shapes appearing in the one-dimensional analysis of the problem in the literature. It also provides a new short and conceptual proof in that case.