Refined regularity at critical points for linear elliptic equations

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 $D^{2}u(x^o)$ 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.