Gespeichert in:
| Hauptverfasser: | , |
|---|---|
| Format: | Preprint |
| Veröffentlicht: |
2026
|
| Schlagworte: | |
| Online-Zugang: | https://arxiv.org/abs/2605.09068 |
| Tags: |
Tag hinzufügen
Keine Tags, Fügen Sie den ersten Tag hinzu!
|
| _version_ | 1866910205489446912 |
|---|---|
| author | Yang, Dong-Hui Guo, Bao-Zhu |
| author_facet | Yang, Dong-Hui Guo, Bao-Zhu |
| contents | In this study, we address the eigenvalue problem given by: \begin{equation*} \begin{cases} -\Div (w\nabla u_i)=\la_iu_i &\text{in } \Om\subset \mathbb{R}^n,\\ u_i=0 &\text{on } \pt \Om, \end{cases} \end{equation*} where $w > 0$ within $\Om$ and $w = 0$ on part of $\partial Ω$. We establish Courant's nodal domain theorem for the corresponding degenerate elliptic differential operator $\mathcal{A}$. Unlike uniformly elliptic operators, degenerate cases often result in the loss of many advantageous properties. Despite this, we show that the essential property that the set $\{ρ\in L^\infty(Ω) \colon \mathcal{A} + ρ\text{ has simple eigenvalues}\}$ forms a residual subset within $(L^\infty(Ω), |\cdot|_\infty)$ still holds for the degenerate elliptic differential operator $\mathcal{A}$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2605_09068 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Some Key Properties of Eigenfunctions Linked to Degenerate Elliptic Differential Operators Yang, Dong-Hui Guo, Bao-Zhu Analysis of PDEs In this study, we address the eigenvalue problem given by: \begin{equation*} \begin{cases} -\Div (w\nabla u_i)=\la_iu_i &\text{in } \Om\subset \mathbb{R}^n,\\ u_i=0 &\text{on } \pt \Om, \end{cases} \end{equation*} where $w > 0$ within $\Om$ and $w = 0$ on part of $\partial Ω$. We establish Courant's nodal domain theorem for the corresponding degenerate elliptic differential operator $\mathcal{A}$. Unlike uniformly elliptic operators, degenerate cases often result in the loss of many advantageous properties. Despite this, we show that the essential property that the set $\{ρ\in L^\infty(Ω) \colon \mathcal{A} + ρ\text{ has simple eigenvalues}\}$ forms a residual subset within $(L^\infty(Ω), |\cdot|_\infty)$ still holds for the degenerate elliptic differential operator $\mathcal{A}$. |
| title | Some Key Properties of Eigenfunctions Linked to Degenerate Elliptic Differential Operators |
| topic | Analysis of PDEs |
| url | https://arxiv.org/abs/2605.09068 |