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| Main Author: | |
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| Format: | Preprint |
| Published: |
2024
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2411.18852 |
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| _version_ | 1866912136328904704 |
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| author | Xu, Ziyi |
| author_facet | Xu, Ziyi |
| contents | We investigate the $W^{1,p}$ estimates of the Neumann problem for the Schrödinger equation $-Δu+ V u={\rm div}(f)$ in the region above a convex graph. For any $p>2$, we obtain a sufficient condition for the $W^{1,p}$ solvability. As a result, we obtain sharp $W^{1,p}$ estimate $$\|\nabla u\|_{L^p(Ω)}+\|V^\frac{1}{2}u\|_{L^p(Ω)}\leq C\|f\|_{L^p(Ω)}$$
for $1 <p<\infty$ with $d\geq2$ under the assumption that $V$ is a $B_\infty$ weight. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2411_18852 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | $W^{1,p}$ estimates for Schrödinger equation in the region above a convex graph Xu, Ziyi Analysis of PDEs 35J10, 35J25, 35B45 We investigate the $W^{1,p}$ estimates of the Neumann problem for the Schrödinger equation $-Δu+ V u={\rm div}(f)$ in the region above a convex graph. For any $p>2$, we obtain a sufficient condition for the $W^{1,p}$ solvability. As a result, we obtain sharp $W^{1,p}$ estimate $$\|\nabla u\|_{L^p(Ω)}+\|V^\frac{1}{2}u\|_{L^p(Ω)}\leq C\|f\|_{L^p(Ω)}$$ for $1 <p<\infty$ with $d\geq2$ under the assumption that $V$ is a $B_\infty$ weight. |
| title | $W^{1,p}$ estimates for Schrödinger equation in the region above a convex graph |
| topic | Analysis of PDEs 35J10, 35J25, 35B45 |
| url | https://arxiv.org/abs/2411.18852 |