Refined regularity at critical points for linear elliptic 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 $D^{2}u(x^o)$ 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.

31 pages

References in corpus (4)

• On $C^1$, $C^2$, and weak type-$(1,1)$ estimates for linear elliptic operators
• On $C^1$, $C^2$, and weak type-$(1,1)$ estimates for linear elliptic operators: Part II
• Green's function for second order elliptic equations in non-divergence form
• Estimates for Green's functions of elliptic equations in non-divergence form with continuous coefficients