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

Abstract

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

15 pages