Gespeichert in:
| Hauptverfasser: | , , |
|---|---|
| Format: | Preprint |
| Veröffentlicht: |
2025
|
| Schlagworte: | |
| Online-Zugang: | https://arxiv.org/abs/2511.18070 |
| Tags: |
Tag hinzufügen
Keine Tags, Fügen Sie den ersten Tag hinzu!
|
| _version_ | 1866911280903749632 |
|---|---|
| author | Qiu, Yusheng Tan, Jinggang Xia, Aliang |
| author_facet | Qiu, Yusheng Tan, Jinggang Xia, Aliang |
| contents | We investigate the quantitative unique continuation property for solutions to
$$Δ^2_{X} u = V u,$$
where $Δ_{X} = Δ_{x} + |x|^{2β} Δ_{y}$ ($0 < β\leq 1$), with $x \in \mathbb{R}^{m}$ and $y \in \mathbb{R}^{n}$, denotes a class of subelliptic operators of Baouendi-Grushin type. The potential $V$ is assumed to be bounded and satisfy $|Z V| \leq K ψ$ for some constant $K>0$,
where $Z= \sum_{i=1}^m x_i \partial_{x_i} + (β+1)\sum_{j=1}^n y_j \partial_{y_j}$, $ψ$ is the angle function given by $ψ= \frac{|x|^{2β}}{ρ^{2β}}$,
and $$ρ(x,y) = \left(|x|^{2(β+1)} + (β+1)^2 |y|^2\right)^{\frac{1}{2(β+1)}}$$ defines the associated pseudo-gauge. By adapting Almgren's approach, we establish an almost monotonicity formula for the frequency function. As a consequence, we derive a quantitative unique continuation result for solutions to the fourth-order subelliptic equation. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2511_18070 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Quantitative unique continuation property for fourth-order Baouendi-Grushin type subelliptic operators with a potential Qiu, Yusheng Tan, Jinggang Xia, Aliang Analysis of PDEs We investigate the quantitative unique continuation property for solutions to $$Δ^2_{X} u = V u,$$ where $Δ_{X} = Δ_{x} + |x|^{2β} Δ_{y}$ ($0 < β\leq 1$), with $x \in \mathbb{R}^{m}$ and $y \in \mathbb{R}^{n}$, denotes a class of subelliptic operators of Baouendi-Grushin type. The potential $V$ is assumed to be bounded and satisfy $|Z V| \leq K ψ$ for some constant $K>0$, where $Z= \sum_{i=1}^m x_i \partial_{x_i} + (β+1)\sum_{j=1}^n y_j \partial_{y_j}$, $ψ$ is the angle function given by $ψ= \frac{|x|^{2β}}{ρ^{2β}}$, and $$ρ(x,y) = \left(|x|^{2(β+1)} + (β+1)^2 |y|^2\right)^{\frac{1}{2(β+1)}}$$ defines the associated pseudo-gauge. By adapting Almgren's approach, we establish an almost monotonicity formula for the frequency function. As a consequence, we derive a quantitative unique continuation result for solutions to the fourth-order subelliptic equation. |
| title | Quantitative unique continuation property for fourth-order Baouendi-Grushin type subelliptic operators with a potential |
| topic | Analysis of PDEs |
| url | https://arxiv.org/abs/2511.18070 |