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 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 ∂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 [B. Bogosel, T. Giletti, and A. Tellini, Nonlinear Analysis 213 (2021) 112528] for its connection with road-field models. Building on the considerations of [B. Bogosel, T. Giletti, and A. Tellini, Nonlinear Analysis 213 (2021) 112528], we have two main objectives here: first, investigate the question of optimal resources distribution inside the domain Ω and on the surface ∂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, Ann. Sc. Norm. Super. Pisa Cl. Sci. 14 (2015) 1025-1063; J. J. Langford, Canad. J. Math. 75 (2023) 108-139]. 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.