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Following recent interest in the qualitative analysis of some optimal controland shape optimisation problems, we provide in this article a detailed study ofthe optimisation of Robin boundary conditions in PDE constrained calculus ofvariations. Our main model consists of an elliptic PDE of the form $-\Deltau_\beta=f(x,u_\beta)$ endowed with the Robin boundary conditions $\partial_\nuu_\beta+\beta(x)u_\beta=0$. The optimisation variable is the function $\beta$,which is assumed to take values between 0 and 1 and to have a fixed integral.Two types of criteria are under consideration: the first one is non-energeticcriteria. In other words, we aim at optimising functionals of the form$\mathcal J(\beta)=\int_{\Omega\text{ or }\partial \Omega}j(u_\beta)$. We provethat, depending on the monotonicity of the function $j$, the optimisers may beof \emph{bang-bang} type (in other words, the optimisers write $1_\Gamma$ forsome measurable subset $\Gamma$ of $\partial \ Omega $) or, on the contrary,that they may only take values strictly between 0 and 1. This has consequencefor a related shape optimisation problem, in which one tries to find where onthe boundary Neumann ($\partial_\nu u=0$ ) and constant Robin conditions($\partial_\nu u+u=0$) should be placed in order to optimise criteria. Theproofs for this first case rely on new fine oscillatory techniques, used incombination with optimality conditions. We then investigate the case ofcompliance-type functionals. For such energetic functionals, we give anin-depth analysis and even some explicit characterisation of optimal $\beta^*$.

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