Unique continuation for a gradient inequality with $L^n$ potential
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arXiv
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| Auteurs principaux: | , , |
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| Format: | Preprint |
| Publié: |
2024
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| _version_ | 1866910925264519168 |
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| author | Coffman, Adam Pan, Yifei Zhang, Yuan |
| author_facet | Coffman, Adam Pan, Yifei Zhang, Yuan |
| contents | We establish a unique continuation property for solutions of the differential inequality $|\nabla u|\leq V|u|$, where $V$ is locally $L^n$ integrable on a domain in $\mathbb R^n$. A stronger uniqueness result is obtained if in addition the solutions are locally Lipschitz. One application is a finite order vanishing property in the $L^2$ sense for the exponential of $W^{1,n}$ functions. We further discuss related results for the Cauchy-Riemann operator $\bar\partial$ and characterize the vanishing order for smooth extension of holomorphic functions across the boundary. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2402_15503 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Unique continuation for a gradient inequality with $L^n$ potential Coffman, Adam Pan, Yifei Zhang, Yuan Analysis of PDEs Complex Variables 35R45, 26B35, 32A40, 35A02 We establish a unique continuation property for solutions of the differential inequality $|\nabla u|\leq V|u|$, where $V$ is locally $L^n$ integrable on a domain in $\mathbb R^n$. A stronger uniqueness result is obtained if in addition the solutions are locally Lipschitz. One application is a finite order vanishing property in the $L^2$ sense for the exponential of $W^{1,n}$ functions. We further discuss related results for the Cauchy-Riemann operator $\bar\partial$ and characterize the vanishing order for smooth extension of holomorphic functions across the boundary. |
| title | Unique continuation for a gradient inequality with $L^n$ potential |
| topic | Analysis of PDEs Complex Variables 35R45, 26B35, 32A40, 35A02 |
| url | https://arxiv.org/abs/2402.15503 |