A note on the rearrangement of functions in time and on the parabolic Talenti inequality

TL;DR

This paper investigates the possibility of rearranging the source term in both space and time for parabolic PDEs to increase solution concentration, concluding that such a maximal rearrangement does not exist.

Contribution

It proves that rearranging the source term in both space and time cannot produce a maximal element in the concentration ordering for parabolic equations.

Findings

Rearranging in space alone increases solution concentration.

Rearranging in both space and time does not admit a maximal element.

The classical spatial rearrangement property does not extend to space-time rearrangements.

Abstract

Talenti inequalities are a central feature in the qualitative analysis of PDE constrained optimal control as well as in calculus of variations. The classical parabolic Talenti inequality states that if we consider the parabolic equation ${\frac{\partial u}{\partial t}}-\Delta u=f=f(t,x)$ then, replacing, for any time $t$, $f(t,\cdot)$ with its Schwarz rearrangement $f^\#(t,\cdot)$ increases the concentration of the solution in the following sense: letting $v$ be the solution of ${\frac{\partial v}{\partial t}}-\Delta v=f^\#$ in the ball, then the solution $u$ is less concentrated than $v$. This property can be rephrased in terms of the existence of a maximal element for a certain order relationship. It is natural to try and rearrange the source term not only in space but also in time, and thus to investigate the existence of such a maximal element when we rearrange the function with respect to the two variables. In the present paper we prove that this is not possible.