Uniqueness of solutions to an elliptic inequality with rapid decay at infinity
Fuente:
arXiv
Salvato in:
| Autori principali: | , , |
|---|---|
| Natura: | Preprint |
| Pubblicazione: |
2025
|
| Soggetti: | |
| Accesso online: | |
| Tags: |
Aggiungi Tag
Nessun Tag, puoi essere il primo ad aggiungerne!!
|
| _version_ | 1866916746440474624 |
|---|---|
| author | Golgeleyen, F. Imanuvilov, O. Y. Yamamoto, M. |
| author_facet | Golgeleyen, F. Imanuvilov, O. Y. Yamamoto, M. |
| contents | We consider an elliptic differential inequality: $\vert Δu(x) \vert \le C_0(\YYYY^{-γ}\vert u(x)\vert + \YYYY^{-θ}\vert \nabla u(x)\vert)$ in an exterior domain $\R^n \setminus \ooo{U}$, where $U$ is a simply connected bounded domain $U$, $x := (y,z) \in \R^n$ with $y \in \R^m$ and $z\in \R^{n-m}$ for given $m\in \{ 1, ..., n\}$, and $γ, θ\in \R$ are constants. We assume that $u(x)$ decays with exponential rate in the $y$-coordinates and polynomial rate in the $z$-coordinates as $\vert x\vert \to \infty$. We prove that if decay rates of $u$ satisfy certain conditions related to the constants $γ, θ\in \R$, then $u\equiv 0$ in $\UUUUU$. The key is a Carleman estimate with typical cut-off arguments. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2505_14431 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Uniqueness of solutions to an elliptic inequality with rapid decay at infinity Golgeleyen, F. Imanuvilov, O. Y. Yamamoto, M. Analysis of PDEs 35R30 We consider an elliptic differential inequality: $\vert Δu(x) \vert \le C_0(\YYYY^{-γ}\vert u(x)\vert + \YYYY^{-θ}\vert \nabla u(x)\vert)$ in an exterior domain $\R^n \setminus \ooo{U}$, where $U$ is a simply connected bounded domain $U$, $x := (y,z) \in \R^n$ with $y \in \R^m$ and $z\in \R^{n-m}$ for given $m\in \{ 1, ..., n\}$, and $γ, θ\in \R$ are constants. We assume that $u(x)$ decays with exponential rate in the $y$-coordinates and polynomial rate in the $z$-coordinates as $\vert x\vert \to \infty$. We prove that if decay rates of $u$ satisfy certain conditions related to the constants $γ, θ\in \R$, then $u\equiv 0$ in $\UUUUU$. The key is a Carleman estimate with typical cut-off arguments. |
| title | Uniqueness of solutions to an elliptic inequality with rapid decay at infinity |
| topic | Analysis of PDEs 35R30 |
| url | https://arxiv.org/abs/2505.14431 |