A self-contained proof of the variational characterization of the principal eigenvalue of $-Δ+q$ under general Robin boundary conditions of non-classical type

In this paper, we are going to deliver a completely self-contained proof of the variational characterization of the principal eigenvalue of $-Δ+q$ in $Ω$ under Robin boundary conditions of non-classical type, ∂ u/∂ n+β(x)u=0 on ∂Ω, where $β(x)$ is allowed to change sign on ∂Ω. The classical case occurs in the very special case when β≥ 0 on ∂Ω.