Spectrum properties of mixed operators under the mixed boundary conditions
Lovelesh Sharma
TL;DR
This work develops a rigorous spectral theory for a mixed local–nonlocal operator L = -Δ + (-Δ)^s under mixed Dirichlet/Neumann boundary conditions on a bounded domain. Using a variational framework in the energy space X^{1,2}_{ m D}(U) with energy η(u)^2, the authors establish a Poincaré-type inequality, well-posed weak formulations, and compact embeddings that yield the existence of eigenpairs. They prove the first eigenvalue λ1 > 0 and simple, and construct a full spectrum {λk} → ∞ with corresponding eigenfunctions {ek} that form orthonormal bases, providing both min-max and Rayleigh quotient characterizations. These results lay a foundational spectral framework for mixed local–nonlocal operators under mixed boundary conditions, enabling further analysis of nonlinear and applied problems in ecology, diffusion, and beyond.
Abstract
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\: &= λu,~~\text{in} ~Ω, u&=0~~~~~\text{in} ~~{U^c}, \mathcal{N}_s(u)&=0 ~~~~~\text{in} ~~{\mathcal{N}}, \frac{\partial u}{\partial ν}&=0 ~~~~~\text{in}~~ \partial Ω\cap \overline{\mathcal{N}}, \end{split} \right.\tag{$P_λ$} \end{equation} where $U= (Ω\cup {\mathcal{N}} \cup (\partialΩ\cap\overline{\mathcal{N}}))$, $Ω\subseteq \mathbb{R}^n$ is a non empty bounded open set with sufficiently smooth boundary $\partialΩ$, say of class $C^1$, and $\mathcal{D}$, $\mathcal{N}$ are open subsets of $\mathbb{R}^n\setminus{\bar{Ω}}$ such that $\overline{\mathcal{D} \cup {\mathcal{N}}}= \mathbb{R}^n\setminusΩ$, $\mathcal{D} \cap {\mathcal{N}}= \emptyset $ and $Ω\cup \mathcal{N}$ is a bounded set with sufficiently smooth boundary, $λ>0$ is a real parameter and $\mathcal{L}= -Δ+(-Δ)^{s},~ \text{for}~s \in (0, 1).$