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

In this paper we address 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 [9] for its connection with road-field models. Building on the considerations of [9], 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 Langford [34, 35]. 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?