On the eigenvalue problem for a bulk/surface elliptic system

Abstract

The paper addresses the doubly elliptic eigenvalue problem $$\{\begin{array}{ll}-\hat{I}”u=\hat{I}»u& \text{in I^c◯,}\\ u=0& \text{on I^“0,}\\ -\hat{I}{”}_{\hat{I}}“u+{\partial }_{\hat{I}}½u=\hat{I}»u& \text{on I^“1,}\end{array}$$ where $\hat{I}\text{c◯}$ is a bounded open subset of ${\mathbb{R}}^N$ ($N\ge 2$) with $C^1$ boundary $\hat{I}“=\hat{I}{“}_0\cup \hat{I}{“}_1$, $\hat{I}{“}_0\cap \hat{I}{“}_1=\varnothing$, $\hat{I}{“}_1$ being nonempty and relatively open on $\hat{I}“$. Moreover ${\mathcal{H}}^{N-1}(\overline{\hat{I}}{“}_0\cap \overline{\hat{I}}{“}_1)=0$ and ${\mathcal{H}}^{N-1}(\hat{I}{“}_0)>0$. We recognize that $L^2(\hat{I}\text{c◯})\times L^2(\hat{I}{“}_1)$ admits a Hilbert basis of eigenfunctions of the problem and we describe the eigenvalues. Moreover, when $\hat{I}“$ is at least $C^2$ and $\overline{\hat{I}}{“}_0\cap \overline{\hat{I}}{“}_1=\varnothing$, we give several qualitative properties of the eigenfunctions.