Analytic properties of heat equation solutions and reachable sets

TL;DR

This paper explores the analytic extension of heat equation solutions in multiple dimensions, characterizing their reachable sets and providing sharp domains of analyticity, with special results for the case of a ball.

Contribution

It generalizes known one-dimensional results to higher dimensions, establishing sharp analyticity domains and their relation to reachable functions for the heat equation.

Findings

Solutions are analytically extendable to a specific domain rom the domain or any bounded Lipschitz domain.

The domain or a ball is sharp, with no larger domain allowing the same extension.

Functions analytic in a neighborhood of or a ball are reachable via the heat equation.

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.