On the Schrödinger equations with $B_{\infty }$ potentials in the region above a Lipschitz graph

Abstract

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 $$1<p<2+\epsilon .$$ Moreover, we also establish the $W^{1,p}$ estimate for Neumann problem for $$\frac{3}{2}-\epsilon <p<3+\epsilon .$$ As a by-product, we also obtain that the $L^p$ Neumann problem is uniquely solvable for $1<p<2+\epsilon .$ The only previously known estimates of this type pertain to the classical Schrödinger equation $-\Delta u+Vu=0$ in $\Omega$ and $\frac{\partial u}{\partial n}=g$ on $\partial \Omega$ 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 $1<p\le 2$. All the ranges of $p$ are sharp.