Unique continuation of Schrödinger-type equations for $\bar\partial$ II
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866909224702836736 |
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| author | Pan, Yifei Zhang, Yuan |
| author_facet | Pan, Yifei Zhang, Yuan |
| contents | In this paper, we extend our earlier unique continuation results \cite{PZ2} for the Schrödinger-type inequality $ |\bar\partial u| \le V|u|$ on a domain in $\mathbb C^n$ by removing the smoothness assumption on solutions $u = (u_1, \ldots, u_N)$. More specifically, we establish the unique continuation property for $W_{loc}^{1,1}$ solutions when the potential $V\in L_{loc}^p $, $ p>2n$; and for $W_{loc}^{1,2n+ε}$ solutions when $V\in L_{loc}^{2n}$ with $N=1$ or $n = 2$. Although the unique continuation property fails in general if $V\in L_{loc}^{p}, p<2n$, we show that the property still holds for $W_{loc}^{1,1}$ solutions when $V $ is a small constant multiple of $ \frac{1}{|z|}$. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2406_10749 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Unique continuation of Schrödinger-type equations for $\bar\partial$ II Pan, Yifei Zhang, Yuan Complex Variables Analysis of PDEs Primary 32W05, Secondary 35J10 In this paper, we extend our earlier unique continuation results \cite{PZ2} for the Schrödinger-type inequality $ |\bar\partial u| \le V|u|$ on a domain in $\mathbb C^n$ by removing the smoothness assumption on solutions $u = (u_1, \ldots, u_N)$. More specifically, we establish the unique continuation property for $W_{loc}^{1,1}$ solutions when the potential $V\in L_{loc}^p $, $ p>2n$; and for $W_{loc}^{1,2n+ε}$ solutions when $V\in L_{loc}^{2n}$ with $N=1$ or $n = 2$. Although the unique continuation property fails in general if $V\in L_{loc}^{p}, p<2n$, we show that the property still holds for $W_{loc}^{1,1}$ solutions when $V $ is a small constant multiple of $ \frac{1}{|z|}$. |
| title | Unique continuation of Schrödinger-type equations for $\bar\partial$ II |
| topic | Complex Variables Analysis of PDEs Primary 32W05, Secondary 35J10 |
| url | https://arxiv.org/abs/2406.10749 |