Differentiability of Solutions to the Neumann Problem with Low-Regularity Data via Dynamical Systems

TL;DR

This paper establishes conditions under which weak solutions to a second-order elliptic PDE with low-regularity data are differentiable, linking solution regularity to the stability of an associated dynamical system.

Contribution

It introduces new criteria connecting the differentiability of solutions to the stability of a dynamical system derived from the PDE coefficients.

Findings

Solutions are differentiable under square-Dini continuity conditions.

Lipschitz continuity and differentiability at boundary points depend on dynamical system stability.

Provides a framework for analyzing PDE regularity with low-regularity data.

Abstract

We obtain conditions for the differentiability of weak solutions for a second-order uniformly elliptic equation in divergence form with a homogeneous co-normal boundary condition. The modulus of continuity for the coefficients is assumed to satisfy the square-Dini condition and the boundary is assumed to be differentiable with derivatives also having this modulus of continuity. Additional conditions for the solution to be Lipschitz continuous or differentiable at a point on the boundary depend upon the stability of a dynamical system that is derived from the coefficients of the elliptic equation.