A unique continuation property for $|\overline \partial u| \leq V |u|$
Fuente:
arXiv
Salvato in:
| Autore principale: | |
|---|---|
| Natura: | Preprint |
| Pubblicazione: |
2024
|
| Soggetti: | |
| Accesso online: | |
| Tags: |
Aggiungi Tag
Nessun Tag, puoi essere il primo ad aggiungerne!!
|
| _version_ | 1866913387390173184 |
|---|---|
| author | Shi, Ziming |
| author_facet | Shi, Ziming |
| contents | Let $u: Ω\subset \mathbb C^n \to \mathbb C^m$, for $n \geq 2$ and $m \geq 1$. Let $1 \leq p \leq 2$, and $2(2n)^2 -1 \leq q < \infty$ such that $\displaystyle \frac{1}{p} + \frac{1}{p'} = 1$ and $\displaystyle \frac{1}{p} - \frac{1}{p'} = \frac{1}{q}$. Suppose $|\overline \partial u| \leq V |u|$, where $V \in L^q_{\operatorname{loc}}(Ω)$. Then $u$ has a unique continuation property in the following sense: if $u \in W^{1,p}_{\operatorname{loc}}(Ω)$ and for some $z_0 \in Ω$, $\| u \|_{L^{p'}(B(z_0,r))} $ decays faster than any powers of $r$ as $r \to 0$, then $u \equiv 0$. The same result holds for $q=\infty$ if $u$ is scalar-valued ($m=1$). |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2406_07650 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | A unique continuation property for $|\overline \partial u| \leq V |u|$ Shi, Ziming Analysis of PDEs Complex Variables Let $u: Ω\subset \mathbb C^n \to \mathbb C^m$, for $n \geq 2$ and $m \geq 1$. Let $1 \leq p \leq 2$, and $2(2n)^2 -1 \leq q < \infty$ such that $\displaystyle \frac{1}{p} + \frac{1}{p'} = 1$ and $\displaystyle \frac{1}{p} - \frac{1}{p'} = \frac{1}{q}$. Suppose $|\overline \partial u| \leq V |u|$, where $V \in L^q_{\operatorname{loc}}(Ω)$. Then $u$ has a unique continuation property in the following sense: if $u \in W^{1,p}_{\operatorname{loc}}(Ω)$ and for some $z_0 \in Ω$, $\| u \|_{L^{p'}(B(z_0,r))} $ decays faster than any powers of $r$ as $r \to 0$, then $u \equiv 0$. The same result holds for $q=\infty$ if $u$ is scalar-valued ($m=1$). |
| title | A unique continuation property for $|\overline \partial u| \leq V |u|$ |
| topic | Analysis of PDEs Complex Variables |
| url | https://arxiv.org/abs/2406.07650 |