Optimisation of space-time periodic eigenvalues

TL;DR

This paper analyzes how to optimally choose time-space periodic functions to minimize principal eigenvalues of certain operators, with implications for population dynamics, using rearrangement techniques and numerical simulations.

Contribution

It introduces the first comparison results for time rearrangement in parabolic equations and explores Talenti inequalities in this context.

Findings

Time rearrangement can reduce principal eigenvalues in several cases.

Symmetric time rearrangement is beneficial for eigenvalue minimization.

Numerical framework developed for optimizing functions with prescribed rearrangements.

Abstract

The goal of this paper is to provide a qualitative analysis of the optimisation of space-time periodic principal eigenvalues. Namely, considering a fixed time horizon $T$ and the $d$-dimensional torus $\mathbb{T}^d$, let, for any $m\in L^\infty((0,T)\times\mathbb{T}^d)$, $\lambda(m)$ be the principal eigenvalue of the operator $\partial_t-\Delta-m$ endowed with (time-space) periodic boundary conditions. The main question we set out to answer is the following: how to choose $m$ so as to minimise $\lambda(m)$? This question stems from population dynamics. We prove that in several cases it is always beneficial to rearrange $m$ with respect to time in a symmetric way, which is the first comparison result for the rearrangement in time of parabolic equations. Furthermore, we investigate the validity (or lack thereof) of Talenti inequalities for the rearrangement in time of parabolic equations. The numerical simulations which illustrate our results were obtained by developing a framework within which it is possible to optimise criteria with respect to functions having a prescribed rearrangement (or distribution function).