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

TL;DR

This paper analyzes the optimization of Robin boundary conditions in PDEs, revealing conditions under which optimal solutions are bang-bang or smooth, with implications for shape optimization and boundary condition placement.

Contribution

It provides a detailed qualitative analysis of Robin boundary condition optimization, including new oscillatory techniques and explicit characterizations for energetic functionals.

Findings

Optimal Robin boundary conditions can be bang-bang or smooth depending on criteria.

New oscillatory techniques are developed for analyzing optimality.

Explicit characterizations are provided for energetic functional optimizers.

Abstract

Following recent interest in the qualitative analysis of some optimal control and shape optimisation problems, we provide in this article a detailed study of the optimisation of Robin boundary conditions in PDE constrained calculus of variations. Our main model consists of an elliptic PDE of the form $-\Delta u_\beta=f(x,u_\beta)$ endowed with the Robin boundary conditions $\partial_\nu u_\beta+\beta(x)u_\beta=0$. 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-energetic criteria. In other words, we aim at optimising functionals of the form $\mathcal J(\beta)=\int_{\Omega\text{ or }\partial \Omega}j(u_\beta)$. We prove that, depending on the monotonicity of the function $j$, the optimisers may be of \emph{bang-bang} type (in other words, the optimisers write $1_\Gamma$ for some measurable subset $\Gamma$ of $\partial \ Omega $) or, on the contrary, that they may only take values strictly between 0 and 1. This has consequence for a related shape optimisation problem, in which one tries to find where on the boundary Neumann ($\partial_\nu u=0$ ) and constant Robin conditions ($\partial_\nu u+u=0$) should be placed in order to optimise criteria. The proofs for this first case rely on new fine oscillatory techniques, used in combination with optimality conditions. We then investigate the case of compliance-type functionals. For such energetic functionals, we give an in-depth analysis and even some explicit characterisation of optimal $\beta^*$.