Gradient Estimates via Rearrangements for Solutions of Some Schr\"odinger Equations

TL;DR

This paper develops sharp estimates for the gradient of solutions to certain Schr"odinger equations using rearrangement techniques, applicable under weak boundary regularity, and derives related space norm bounds.

Contribution

It introduces a novel application of rearrangement methods to obtain gradient estimates for Schr"odinger equations with minimal boundary regularity assumptions.

Findings

Sharp gradient estimates in Lebesgue and Lorentz spaces

Rearrangement techniques effectively handle weak boundary regularity

Applicable to Dirichlet and Neumann boundary value problems

Abstract

In this article, by applying the well known method for dealing with $p$-Laplace type elliptic boundary value problems, the authors establish a sharp estimate for the decreasing rearrangement of the gradient of solutions to the Dirichlet and the Neumann boundary value problems of a class of Schr\"odinger equations, under the weak regularity assumption on the boundary of domains. As applications, gradient estimates of these solutions in Lebesgue spaces and Lorentz spaces are obtained.