$W^{1,p}$ estimates for Schrödinger equation in the region above a convex graph

We investigate the $W^{1,p}$ estimates of the Neumann problem for the Schrödinger equation $-\mathrm{\Delta }u+Vu=\mathrm{d}\mathrm{i}\mathrm{v}(f)$ 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 $$\mathrm{\parallel }\mathrm{\nabla }u{\mathrm{\parallel }}_{L^p(\mathrm{\Omega })}+\mathrm{\parallel }{V}^{\frac{1}{2}}u{\mathrm{\parallel }}_{L^p(\mathrm{\Omega })}\le C\mathrm{\parallel }f{\mathrm{\parallel }}_{L^p(\mathrm{\Omega })}$$ for $1<p<\mathrm{\infty }$ with $d\ge 2$ under the assumption that $V$ is a $B_{\mathrm{\infty }}$ weight.