On the eigenvalue problem for a bulk/surface elliptic system

The paper addresses the doubly elliptic eigenvalue problem cases -Δ u=λ u &inΩ,\\ u=0 &onΓ_0,\\ -Δ_Γ u +∂_ν u =λ u &onΓ_1, cases where Ω is a bounded open subset of RN (N≥ 2) with C¹ boundary Γ=Γ₀∪Γ₁, Γ₀∩Γ₁=∅, Γ₁ being nonempty and relatively open on Γ. Moreover HN-1(Γ̄₀∩Γ̄₁)=0 and HN-1(Γ₀)>0. We recognize that L²(Ω)× L²(Γ₁) admits a Hilbert basis of eigenfunctions of the problem and we describe the eigenvalues. Moreover, when Γ is at least C² and Γ̄₀∩Γ̄₁=∅, we give several qualitative properties of the eigenfunctions.