On the eigenvalue problem for a bulk/surface elliptic system

The paper addresses the doubly elliptic eigenvalue problem $$\{\begin{array}{ll}-\mathrm{\Delta }u=\lambda u& \text{in }\mathrm{\Omega }\text{,}\\ u=0& \text{on }{\mathrm{\Gamma }}_0\text{,}\\ -{\mathrm{\Delta }}_{\mathrm{\Gamma }}u+{\mathrm{\partial }}_{\nu }u=\lambda u& \text{on }{\mathrm{\Gamma }}_1\text{,}\end{array}$$ where $\mathrm{\Omega }$ is a bounded open subset of ${\mathbb{R}}^N$ ($N\ge 2$) with $C^1$ boundary $\mathrm{\Gamma }={\mathrm{\Gamma }}_0\cup {\mathrm{\Gamma }}_1$, ${\mathrm{\Gamma }}_0\cap {\mathrm{\Gamma }}_1=\mathrm{\varnothing }$, ${\mathrm{\Gamma }}_1$ being nonempty and relatively open on $\mathrm{\Gamma }$. Moreover ${\mathcal{H}}^{N-1}({\bar{\mathrm{\Gamma }}}_0\cap {\bar{\mathrm{\Gamma }}}_1)=0$ and ${\mathcal{H}}^{N-1}({\mathrm{\Gamma }}_0)>0$. We recognize that $L^2(\mathrm{\Omega })\times L^2({\mathrm{\Gamma }}_1)$ admits a Hilbert basis of eigenfunctions of the problem and we describe the eigenvalues. Moreover, when $\mathrm{\Gamma }$ is at least $C^2$ and ${\bar{\mathrm{\Gamma }}}_0\cap {\bar{\mathrm{\Gamma }}}_1=\mathrm{\varnothing }$, we give several qualitative properties of the eigenfunctions.