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We consider a Laplace type problem with a generalized impedance boundarycondition 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 parameter and $x$ is the coordinatealong $\Gamma$. Such problems appear for example in the modelling of smallperturbations 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 studycases where $g$ vanishes at the origin and changes its sign. The main messageis that the well-posedness in the Fredholm sense of the corresponding problemsdepends on the value of $\alpha$. For $\alpha\in[0,1)$, we show that theassociated 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 1Dproblems combined with the derivation of compact embedding results for thefunctional spaces involved in the analysis. The proof of the second resultsrelies on the computation of singularities and the construction of Weyl'ssequences. We also discuss the equivalence between the strong and weakformulations, which is not straightforward. Finally, we provide simplenumerical experiments which seem to corroborate the theorems.

Generalized impedance boundary conditions with vanishing or sign-changing impedance