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On the eigenvalue problem for a bulk/surface elliptic system

The paper addresses the doubly elliptic eigenvalue problem $$\begin{cases} -Δu=λu \qquad &\text{in $Ω$,}\\ u=0 &\text{on $Γ_0$,}\\ -Δ_Γu +\partial_νu =λu\qquad &\text{on $Γ_1$,} \end{cases} $$ where $Ω$ is a bounded open subset of $\mathbb{R}^N$ ($N\ge 2$) with $C^1$ boundary $Γ=Γ_0\cupΓ_1$, $Γ_0\capΓ_1=\emptyset$, $Γ_1$ being nonempty and relatively open on $Γ$. Moreover $\mathcal{H}^{N-1}(\overlineΓ_0\cap\overlineΓ_1)=0$ and $\mathcal{H}^{N-1}(Γ_0)>0$. We recognize that $L^2(Ω)\times L^2(Γ_1)$ admits a Hilbert basis of eigenfunctions of the problem and we describe the eigenvalues. Moreover, when $Γ$ is at least $C^2$ and $\overlineΓ_0\cap\overlineΓ_1=\emptyset$, we give several qualitative properties of the eigenfunctions.