Gilbarg-Serrin Equation and Lipschitz Regularity

TL;DR

This paper investigates conditions under which solutions to uniformly elliptic divergence form equations exhibit Lipschitz or C^1 regularity, revealing cases where regularity holds without Dini continuity of coefficients.

Contribution

It identifies new conditions and examples where Lipschitz or C^1 regularity occurs despite coefficients lacking Dini continuity or mean oscillation.

Findings

Lipschitz and C^1 regularity can hold without Dini continuity.

Existence of weak solutions that are not Lipschitz continuous.

Specific coefficient conditions enabling regularity results.

Abstract

We discuss conditions for Lipschitz and C^1 regularity for a uniformly elliptic equation in divergence form with coefficients that were introduced by Gilbarg & Serrin. In particular, we find cases where Lipschitz or C^1 regularity holds but the coefficients are not Dini continuous, or do not even have Dini mean oscillation. The form of the coefficients also enables us to obtain specific conditions and examples for which there exists a weak solution that is not Lipschitz continuous.