Spectrum properties of mixed operators under the mixed boundary conditions
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arXiv
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| Format: | Preprint |
| Publié: |
2026
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| _version_ | 1866908788461666304 |
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| author | Sharma, Lovelesh |
| author_facet | Sharma, Lovelesh |
| contents | 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
$\mathcal{L}= -Δ+(-Δ)^{s},~ \text{for}~s \in (0, 1).$ |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2601_17878 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Spectrum properties of mixed operators under the mixed boundary conditions Sharma, Lovelesh Analysis of PDEs 47A75, 35J25, 35J20 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 $\mathcal{L}= -Δ+(-Δ)^{s},~ \text{for}~s \in (0, 1).$ |
| title | Spectrum properties of mixed operators under the mixed boundary conditions |
| topic | Analysis of PDEs 47A75, 35J25, 35J20 |
| url | https://arxiv.org/abs/2601.17878 |