On conormal and oblique derivative problem for elliptic equations with Dini mean oscillation coefficients

Abstract

We show that weak solutions to conormal derivative problem for elliptic equations in divergence form are continuously differentiable up to the boundary provided that the mean oscillations of the leading coefficients satisfy the Dini condition, the lower order coefficients satisfy certain suitable conditions, and the boundary is locally represented by a $C^1$ function whose derivatives are Dini continuous. We also prove that strong solutions to oblique derivative problem for elliptic equations in nondivergence form are twice continuously differentiable up to the boundary if the mean oscillations of coefficients satisfy the Dini condition and the boundary is locally represented by a $C^1$ function whose derivatives are double Dini continuous. This in particular extends a result of M. V. Safonov (Comm. Partial Differential Equations 20:1349--1367, 1995)

minor change in a remark; to appear in Indiana University Mathematics Journal

References in corpus (3)

• Elliptic equations in divergence form with partially BMO coefficients
• 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

Cited by in corpus (8)

• Boundary regularity estimates in Hölder spaces with variable exponent
• Schauder type estimates for degenerate or singular elliptic equations with DMO coefficients
• Gradient estimates for singular $p$-Laplace type equations with measure data
• Higher order boundary Harnack principles in Dini type domains
• A simple proof of regularity for $C^{1,\alpha }$ interface transmission problems
• Hessian estimates for non-divergence form elliptic equations arising from composite materials
• $C^2$ estimate for oblique derivative problem with mean Dini coefficients
• Gradient estimates for divergence form parabolic systems