Improvements in $L^2$ Restriction bounds for Neumann Data along closed curves
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arXiv
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| Format: | Preprint |
| Published: |
2022
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| _version_ | 1866916029257482240 |
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| author | Xianchao, Wu |
| author_facet | Xianchao, Wu |
| contents | We seek to improve the restriction bounds of Neumann data of Laplace eigenfunctions $u_h$ by studying the $L^2$ restriction bounds of Neumann data and their $L^2$ concentration as measured by defect measures. Let $γ$ be a closed smooth curve with unit exterior normal $ν$. We can show that $\| h \partial_νu_{h} \|_{L^2(Γ)}=o(1)$ if $\{u_h\}$ is tangentially concentrated with respect to $γ$. As a key ingredient of the proof, we give a detailed analysis of the $L^2$ norms over $γ$ of the Neumann data $h\partial_νu_h$ when mircolocalized away the cotangential direction. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2203_01208 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | Improvements in $L^2$ Restriction bounds for Neumann Data along closed curves Xianchao, Wu Analysis of PDEs We seek to improve the restriction bounds of Neumann data of Laplace eigenfunctions $u_h$ by studying the $L^2$ restriction bounds of Neumann data and their $L^2$ concentration as measured by defect measures. Let $γ$ be a closed smooth curve with unit exterior normal $ν$. We can show that $\| h \partial_νu_{h} \|_{L^2(Γ)}=o(1)$ if $\{u_h\}$ is tangentially concentrated with respect to $γ$. As a key ingredient of the proof, we give a detailed analysis of the $L^2$ norms over $γ$ of the Neumann data $h\partial_νu_h$ when mircolocalized away the cotangential direction. |
| title | Improvements in $L^2$ Restriction bounds for Neumann Data along closed curves |
| topic | Analysis of PDEs |
| url | https://arxiv.org/abs/2203.01208 |