  We consider a Laplace type problem with a generalized impedance boundary
condition of the form $\partial_\nu u=-\partial_x(g\partial_xu)$ on a flat part
$\Gamma$ of the boundary. Here $\nu$ is the outward unit normal vector to
$\partial\Omega$, $g$ is the impedance function and $x$ is the coordinate along
$\Gamma$. Such problems appear for example in the modelling of small
perturbations of the boundary. In the literature, the cases $g=1$ or $g=-1$
have been investigated. In this work, we address situations where $\Gamma$
contains the origin and $g(x)=\mathbb{1}_{x>0}(x)x^\alpha$ or
$g(x)=-\mbox{sign}(x)|x|^\alpha$ with $\alpha\ge0$. In other words, we study
cases where $g$ vanishes at the origin and changes its sign. The main message
is that the well-posedness in the Fredholm sense of the corresponding problems
depends on the value of $\alpha$. For $\alpha\in[0,1)$, we show that the
associated operators are Fredholm of index zero while it is not the case when
$\alpha=1$. The proof of the first results is based on the reformulation as 1D
problems combined with the derivation of compact embedding results for the
functional spaces involved in the analysis. The proof of the second results
relies on the computation of singularities and the construction of Weyl's
sequences. We also discuss the equivalence between the strong and weak
formulations, which is not straightforward. Finally, we provide simple
numerical experiments which seem to corroborate the theorems.
