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

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.