Refined regularity at critical points for linear elliptic equations

2025/06/09 by Choi, Jongkeun, Dong, Hongjie, Kim, Seick
#Analysis of PDEs (math.AP) #FOS: Mathematics

paper · doi:10.48550/arxiv.2506.08281

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 D²u(x^o) exists and satisfies sharp continuity estimates. As a consequence, we obtain ``C^2,α regularity'' at critical points when the coefficients of L are C^α. 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.

Citations

• Regularity of elliptic equations in double divergence form and applications to Green's function estimates
• Geometric regularity for elliptic equations in double-divergence form
• On C¹ C 1 , C² C 2 , and weak type-(1, 1) estimates for linear elliptic operators: part II
• Regularity for quasilinear equations on degenerate singular sets

Related

• Regularity of elliptic equations in double divergence form and applications to Green's function estimates
• Some remarks on the Lp regularity of second derivatives of solutions to\n non-divergence elliptic equations and the Dini condition
• Higher regularity of solutions to fully nonlinear elliptic equations
• Critical regularity for elliptic equations from Littlewood-Paley theory II
• Geometric regularity for elliptic equations in double-divergence form