Refined regularity at critical points for linear elliptic equations

TL;DR

This paper establishes refined regularity results for solutions to linear elliptic equations at critical points, showing existence of second derivatives and sharp continuity estimates under Dini mean oscillation conditions, extending previous theorems.

Contribution

It provides new regularity results at critical points for linear elliptic equations with Dini mean oscillation coefficients, refining prior theorems in the linear case.

Findings

Existence of second derivatives at critical points under Dini mean oscillation.

Sharp continuity estimates for second derivatives at critical points.

Extension of regularity results to non-divergence form equations.

Abstract

We investigate the regularity of solutions to linear elliptic equations in both divergence and non-divergence forms, particularly when the principal coefficients have Dini mean oscillation. We show that if a solution $u$ to a divergence-form equation satisfies $Du(x^o)=0$ at a point, then the second derivative $D^2u(x^o)$ exists and satisfies sharp continuity estimates. As a consequence, we obtain ``$C^{2,\alpha}$ regularity'' at critical points when the coefficients of $L$ are $C^\alpha$. This result refines a theorem of Teixeira (Math. Ann. 358 (2014), no. 1--2, 241--256) in the linear setting, where both linear and nonlinear equations were considered. We also establish an analogous result for equations in non-divergence form.