Boundary determination for the Schrödinger equation with unknown embedded obstacles by local data
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arXiv
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| Autori principali: | , |
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| Natura: | Preprint |
| Pubblicazione: |
2025
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| _version_ | 1866929738987077632 |
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| author | Wu, Chengyu Yang, Jiaqing |
| author_facet | Wu, Chengyu Yang, Jiaqing |
| contents | In this paper, we consider the inverse boundary value problem of the elliptic operator $Δ+q$ in a fixed region $Ω\subset\mathbb{R}^3$ with unknown embedded obstacles $D$. In particular, we give a new and simple proof to uniquely determine $q$ and all of its derivatives at the boundary from the knowledge of the local Dirichlet-to-Neumann map on $\partialΩ$, disregarding the unknown obstacle, where in fact only the local Cauchy data of the fundamental solution is used. Our proof mainly depends on the rigorous singularity analysis on certain singular solutions and the volume potentials of fundamental solution, which is easy to extend to many other cases. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2503_01246 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Boundary determination for the Schrödinger equation with unknown embedded obstacles by local data Wu, Chengyu Yang, Jiaqing Analysis of PDEs In this paper, we consider the inverse boundary value problem of the elliptic operator $Δ+q$ in a fixed region $Ω\subset\mathbb{R}^3$ with unknown embedded obstacles $D$. In particular, we give a new and simple proof to uniquely determine $q$ and all of its derivatives at the boundary from the knowledge of the local Dirichlet-to-Neumann map on $\partialΩ$, disregarding the unknown obstacle, where in fact only the local Cauchy data of the fundamental solution is used. Our proof mainly depends on the rigorous singularity analysis on certain singular solutions and the volume potentials of fundamental solution, which is easy to extend to many other cases. |
| title | Boundary determination for the Schrödinger equation with unknown embedded obstacles by local data |
| topic | Analysis of PDEs |
| url | https://arxiv.org/abs/2503.01246 |