Optimisation of space-time periodic eigenvalues

TL;DR

This work addresses the optimisation of the space-time periodic principal eigenvalue $λ(m)$ for the parabolic operator $∂_t-Δ-m$ on $(0,T)×\mathbb{T}^d$, with two constrained classes for $m$. It proves that optimisers are symmetric in time and explores monotonicity in time under multiplicative potentials, including a detailed Gaussian-potential analysis where time rearrangement is beneficial. The authors develop large- and small-diffusivity asymptotics showing convergence of the optimal controls to a symmetric rearrangement $\bar c$ up to translations, while also demonstrating time-symmetry breaking for certain nonlinear objectives, highlighting limitations of time Talenti-type inequalities. A comprehensive numerical framework is then built to optimise eigenvalues under fixed rearrangements, corroborating the theoretical results and guiding conjectures. Collectively, the paper advances understanding of spectral optimisation for non-selfadjoint parabolic problems and its implications for population dynamics in space-time periodic environments.

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 $m\in L^\infty((0,T)\times\mathbb{T}^d)$, $λ(m)$ be the principal eigenvalue of the operator $\partial_t-Δ-m$ 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 $λ(m)$? 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).

Figures (8)

• Figure 1: Numerical solutions for problem \ref{['Eq:PvHamiltonian']} for various parameters, using bound and average constraints on the function $c$.
• Figure 2: The region $[-\pi,\pi]\times [-1,1]$ containing the graph of the cosine function is divided into $N\times K$ rectangles. The covered area in each small rectangle is indicated in the figure. The area of a small rectangle is $(\Delta t)\times (\Delta c) \approx 0.126$. A function having the same discrete rearrangement will be characterised by a $K\times N$ matrix with entries decreasing on columns, having fixed sum on lines (rightmost column above).
• Figure 3: An example of a discrete function defined on the $N\times K$ rectangular grid having the required properties to be a discrete rearrangement of the cosine function: sum of values of lines equals to the same quantity for the cosine, values are decreasing on vertical columns. The area of a small rectangle is $(\Delta t)\times (\Delta c) \approx 0.126$.
• Figure 4: An alternative illustration for the discrete rearrangement of the cosine illustrated in Figure \ref{['fig:rearrangement-arb']}. In each one of the $N\times K$ cells a rectangle is plotted having area proportional to the area covered by the rearrangement in that cell.
• Figure 5: Numerical optimisation for the functional \ref{['Eq:PvHamiltonian']} for functions $c$ having a prescribed rearrangement. Numerical results show that there is a different behaviour of the numerical optimiser with respect to the parameter $p$. For $p=2$ the solution is symmetric, while for $p=6$ the symmetry is lost. This confirms the results of Theorem \ref{['Th:SymmetryBreaking']}.

Theorems & Definitions (49)

• Definition 1: Symmetric decreasing rearrangement
• Lemma 2
• Remark 3
• Remark 4
• Remark 5
• Theorem I
• Proposition 6
• Theorem II
• Theorem III
• Theorem IV