Saved in:
| Main Authors: | , |
|---|---|
| Format: | Preprint |
| Published: |
2024
|
| Subjects: | |
| Online Access: | https://arxiv.org/abs/2411.18458 |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1866910718791516160 |
|---|---|
| author | Geng, Jun Xu, Ziyi |
| author_facet | Geng, Jun Xu, Ziyi |
| contents | In this paper we investigate the $L^p$ regularity, $L^p$ Neumann and $W^{1,p}$ problems for generalized Schrödinger operator $-\text{div}(A\nabla )+ V $ in the region above a Lipschitz graph under the assumption that $A$ is elliptic, symmetric and $x_d-$independent. Specifically, we prove that the $L^p$ regularity problem is uniquely solvable for $$1<p<2+\varepsilon.$$
Moreover, we also establish the $W^{1,p}$ estimate for Neumann problem for
$$\frac{3}{2}-\varepsilon<p<3+\varepsilon.$$
As a by-product, we also obtain that the $L^p$ Neumann problem is uniquely solvable for $1<p<2+\varepsilon.$
The only previously known estimates of this type pertain to the classical Schrödinger equation $-Δu+ Vu=0$ in $Ω$ and $\frac{\partial u}{\partial n}=g$ on $\partialΩ$ which was obtained by Shen [Z. Shen, On the Neumann problem for Schrödinger operators in Lipschitz domains, Indiana Univ. Math. J. 43 (1994)] for ranges $1<p\leq 2$.
All the ranges of $p$ are sharp. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2411_18458 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | On the Schrödinger equations with $B_\infty$ potentials in the region above a Lipschitz graph Geng, Jun Xu, Ziyi Analysis of PDEs 35J10, 35J25, 35B45 In this paper we investigate the $L^p$ regularity, $L^p$ Neumann and $W^{1,p}$ problems for generalized Schrödinger operator $-\text{div}(A\nabla )+ V $ in the region above a Lipschitz graph under the assumption that $A$ is elliptic, symmetric and $x_d-$independent. Specifically, we prove that the $L^p$ regularity problem is uniquely solvable for $$1<p<2+\varepsilon.$$ Moreover, we also establish the $W^{1,p}$ estimate for Neumann problem for $$\frac{3}{2}-\varepsilon<p<3+\varepsilon.$$ As a by-product, we also obtain that the $L^p$ Neumann problem is uniquely solvable for $1<p<2+\varepsilon.$ The only previously known estimates of this type pertain to the classical Schrödinger equation $-Δu+ Vu=0$ in $Ω$ and $\frac{\partial u}{\partial n}=g$ on $\partialΩ$ which was obtained by Shen [Z. Shen, On the Neumann problem for Schrödinger operators in Lipschitz domains, Indiana Univ. Math. J. 43 (1994)] for ranges $1<p\leq 2$. All the ranges of $p$ are sharp. |
| title | On the Schrödinger equations with $B_\infty$ potentials in the region above a Lipschitz graph |
| topic | Analysis of PDEs 35J10, 35J25, 35B45 |
| url | https://arxiv.org/abs/2411.18458 |