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

TL;DR

This paper establishes $W^{1,p}$ estimates for the Neumann problem of the Schr"odinger equation in regions above convex graphs, providing conditions for solvability and sharp estimates involving potential weights.

Contribution

It offers new sufficient conditions for $W^{1,p}$ solvability of the Schr"odinger equation in convex regions, extending previous results to weighted potentials and a broad range of p.

Findings

Established $W^{1,p}$ solvability conditions for $p>2$

Derived sharp $W^{1,p}$ estimates involving $V^{1/2}u$

Extended results to weights $V$ in $B__ ext{infty}$ class

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.