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| Main Authors: | , |
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| Format: | Preprint |
| Published: |
2024
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2411.18458 |
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Table of 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.