$W^{1,p}$ estimates for Schr\"odinger equation in the region above a convex graph

Abstract

We investigate the $W^{1,p}$ estimates of the Neumann problem for the Schr\"odinger equation $-\Delta u+ V u={\rm div}(f)$ 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 $$\|\nabla u\|_{L^p(\Omega)}+\|V^\frac{1}{2}u\|_{L^p(\Omega)}\leq C\|f\|_{L^p(\Omega)}$$ for $1 <p<\infty$ with $d\geq2$ under the assumption that $V$ is a $B_\infty$ weight.