A discrete-time overdetermined problem for the heat equation

TL;DR

This paper investigates a discrete-time overdetermined problem for the heat equation, establishing that solutions exist only for spherical domains and exploring the interplay of geometric and spectral properties.

Contribution

It introduces a novel discrete-time overdetermined problem for the heat equation and proves rigidity results linking domain shape to overdetermined conditions.

Findings

Solutions exist only if the domain is a ball.

The overdetermined condition captures geometric or spectral information.

The methods extend to Riemannian manifolds.

Abstract

In this paper, we consider a parabolic counterpart of Serrin's overdetermined problem, in which the overdetermined condition (constant flux condition) is imposed only on a discrete infinite set of time values. We show that, under suitable regularity assumptions on the domain, such a discrete-time overdetermined problem admits a solution if and only if the domain is a ball. Remarkably, depending on the temporal scale, the same overdetermined condition captures either geometric or spectral information, yet both mechanisms lead to the same rigidity conclusion. We study both the case in which the constant flux condition is imposed on the boundary and the case in which the constant flux condition is imposed on an interior surface. We remark that the methods employed in our analysis do not depend on the location of the overdetermined surface but only on whether the sequence of time instants accumulates away from zero. Finally, we will show how this problem generalizes to complete Riemannian manifolds.