Differentiability of Solutions to Second-Order Elliptic Equations via Dynamical Systems

TL;DR

This paper studies how the regularity of solutions to second-order elliptic equations depends on the coefficients' continuity, extending previous results by using dynamical systems to analyze solution behavior.

Contribution

It introduces new conditions on coefficients with square-Dini continuity that ensure solutions are Lipschitz or differentiable, extending prior Dini condition results.

Findings

Solutions are Lipschitz continuous under square-Dini conditions.

Solutions are differentiable at a point with additional coefficient conditions.

The method uses asymptotic analysis of a derived dynamical system.

Abstract

For a second-order elliptic equation in divergence form we investigate conditions on the coefficients which imply that all solutions 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 that examples show are sharp. Our results extend those of previous authors who assume the modulus of continuity satisfies the Dini condition. Our method involves the study of asymptotic properties of solutions to a dynamical system that is derived from the coefficients of the elliptic equation.