Second-order differentiability for solutions of elliptic equations in the plane

TL;DR

This paper studies conditions on the coefficients of second-order elliptic equations in the plane that ensure solutions have Lipschitz continuous or differentiable derivatives, extending previous results beyond the Dini condition.

Contribution

It introduces new conditions related to a dynamical system derived from the coefficients, extending second-order differentiability results beyond the Dini continuity assumption.

Findings

Solutions have Lipschitz continuous derivatives under square-Dini conditions.

Additional conditions from a dynamical system imply differentiability at a point.

Extends previous work from Dini to square-Dini continuity.

Abstract

For a second-order elliptic equation of nondivergence form in the plane, we investigate conditions on the coefficients which imply that all strong solutions have first-order derivatives that are Lipschitz continuous or differentiable at a given point. We assume the coefficients have modulus of continuity satisfying the square-Dini condition, and obtain additional conditions associated with a dynamical system that is derived from the coefficients of the elliptic equation. Our results extend those of previous authors who assume the modulus of continuity satisfies the Dini condition.