Some optimal control and shape optimisation problems for bulk-surface cooperative systems
Andrea Gentile, Idriss Mazari-Fouquer, Raphaël Prunier
TL;DR
The article advances the theory of bulk–surface cooperative PDE systems by establishing Talenti-type comparison and rigidity results for an optimally controlled coupled interior–boundary problem, and by conducting a detailed shape-optimization analysis with constant resource distributions. It introduces cap symmetrisation techniques to compare the coupled system with symmetrised data and proves rigidity under $L^p$-norm equality, extending Langford’s Robin-problem framework to a two-species setting. In the shape-optimisation part, it shows nonexistence of a global minimiser under volume constraints in parameter regimes where boundary resources dominate interior growth, and provides a rigorous second-order ball analysis that yields local minimality in the interior-dominant regime (for $2\le d\le 5$) while indicating non-minimality in other regimes. Overall, the work connects Talenti inequalities, energy-based optimal control, and spectral shape optimisation in a bulk–surface context, with implications for design problems in road-field and cell-polarisation models.
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 $Ω$ be a fixed domain, we assume that a population (with density $u$) lives inside $Ω$ and can access some resources $f$, while a second population (with density $v$) lives on the boundary $\partial Ω$ 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 $Ω$ and on the surface $\partial Ω$, 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 $Ω$ 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 $Ω$ 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.