Spectrum properties of mixed operators under the mixed boundary conditions

In this paper, we describe the spectrum properties of mixed operators, precisely the superposition of the classical Laplace operator and the fractional Laplace operator in the presence of mixed boundary conditions, that is \begin{equation} \label{1} \left\{\begin{split} \mathcal{L}u\: &= \lambda u,~~\text{in} ~\Omega, u&=0~~~~~\text{in} ~~{U^c}, \mathcal{N}_s(u)&=0 ~~~~~\text{in} ~~{\mathcal{N}}, \frac{\partial u}{\partial \nu}&=0 ~~~~~\text{in}~~ \partial \Omega \cap \overline{\mathcal{N}}, \end{split} \right.\tag{$P_\lambda$} \end{equation} where $U= (\Omega \cup {\mathcal{N}} \cup (\partial\Omega\cap\overline{\mathcal{N}}))$, $\Omega \subseteq \mathbb{R}^n$ is a non empty bounded open set with sufficiently smooth boundary $\partial\Omega$, say of class $C^1$, and $\mathcal{D}$, $\mathcal{N}$ are open subsets of $\mathbb{R}^n\setminus{\bar{\Omega }}$ such that $\overline{{\mathcal{D}} \cup {\mathcal{N}}}= \mathbb{R}^n\setminus{\Omega}$, $\mathcal{D} \cap {\mathcal{N}}= \emptyset$ and $\Omega\cup \mathcal{N}$ is a bounded set with sufficiently smooth boundary, $\lambda >0$ is a real parameter and $\mathcal{L}= -\Delta+(-\Delta)^{s},~ \text{for}~s \in (0, 1).$