Some optimal control and shape optimisation problems for bulk-surface cooperative systems

TL;DR

This paper studies optimal resource distribution and shape optimization for a coupled bulk-surface cooperative PDE system modeling two interacting populations, providing inequalities, comparison results, and shape analysis for survival maximization.

Contribution

It introduces new inequalities and shape analysis techniques for a coupled PDE system, extending previous symmetrization results and exploring optimal shapes for species survival.

Findings

Rigid Talenti inequalities for ball domains

Comparison results for resource distribution

Analysis of shape optimality and non-existence of optimal shapes in certain regimes

Abstract

The goal of this paper is to address some optimal control and shape optimisation problems arising from bulk-surface cooperative systems. The basic model under consideration is the following: letting $\Omega$ be a fixed domain, we assume that a population (with density $u$) lives inside $\Omega$ and can access some resources $f$, while a second population (with density $v$) lives on the boundary $\partial \Omega$ and can access other resources $g$. These two populations are coupled in a cooperative manner by a constant exchange rate at the boundary, leading to a non-standard PDE system that has already been studied in previous works by Bogosel, Giletti and Tellini, for its connection with road-field models. Building on the considerations of the aforementioned previous works, we have two main objectives here: first, investigate the question of optimal resources distribution inside the domain $\Omega$ and on the surface $\partial \Omega$, i.e. how to spread resources in order to guarantee an optimal survival of the two species. We establish rigid Talenti inequalities and comparison results when $\Omega$ is a ball, extending in particular the results of J. J. Langford on symmetrisation for Neumann and Robin problems. Second, when the resources distribution $f$ and $g$ are constant, we provide a partial analysis of the natural shape optimisation problem: which shape $\Omega$ maximises the survival rate of the two species? Namely, we show that in certain regimes there can be no optimal shape and, by computing second-order shape derivatives, we investigate the local optimality of the ball.