Global Boundedness of the Gradient for a Class of Schr\"odinger Equations

TL;DR

This paper establishes the global boundedness of the gradient for solutions to certain Schr"odinger equations with minimal assumptions, extending results to arbitrary bounded semi-convex domains.

Contribution

It applies a method by Cianchi and Maz'ya to prove gradient boundedness under minimal data and boundary regularity assumptions, including semi-convex domains.

Findings

Gradient of solutions is globally bounded under minimal assumptions.

Results apply to both Dirichlet and Neumann boundary conditions.

Extends to arbitrary bounded semi-convex domains.

Abstract

In this paper, via applying the method developed by A. Cianchi and V. Maz'ya, the author obtains the global boundedness of the gradient for solutions to Dirichlet and Neumann problems of a class of Schr\"odinger equations under the minimal assumptions for integrability on the data and regularity on the boundary of the domain. Moreover, the case of arbitrary bounded semi-convex domains is also considered.