Gradient Estimate for Solutions of Second-Order Elliptic Equations

TL;DR

This paper derives a sharp local gradient estimate for solutions of second-order elliptic equations with coefficients that are square-Dini continuous at a point, even when solutions are not Lipschitz continuous there.

Contribution

It provides a novel sharp gradient estimate for elliptic equations with square-Dini continuous coefficients at a point, extending previous regularity results.

Findings

Established a sharp local gradient estimate at a point.

Handled solutions lacking Lipschitz continuity at the point.

Extended regularity theory for elliptic equations with specific coefficient continuity.

Abstract

We obtain a local estimate for the gradient of solutions to a second-order elliptic equation in divergence form with bounded measurable coefficients that are square-Dini continuous at the single point x=0. In particular, we treat the case of solutions that are not Lipschitz continuous at x=0. We show that our estimate is sharp.