Qualitative analysis of optimisation problems with respect to non-constant Robin coefficients

Following recent interest in the qualitative analysis of some optimal control and shape optimisation problems, we provide this article a detailed study Robin boundary conditions PDE constrained calculus variations. Our main model consists an elliptic form −∆u β = f (x, u β) endowed with ∂ν +β(x)u 0. The variable is function β, which assumed to take values between 0 1 have fixed integral. Two types criteria are under consideration: first one non-energetic criteria. In other words, aim at optimising functionals J (β) =\int_{Ω or ∂Ω} j(u β). We prove that, depending on monotonicity j, optimisers may be bang-bang type (in write 1Γ for measurable subset Γ ∂Ω) or, contrary, that they only strictly 1. This has consequence related problem, tries find where Neumann (∂ν 0) constant + should placed order optimise proofs case rely new fine oscillatory techniques, used combination optimality conditions. then investigate compliance-type functionals. For such energetic functionals, give in-depth even explicit characterisation * .