On $C^1$, $C^2$, and weak type-$(1,1)$ estimates for linear elliptic operators

Abstract

We show that any weak solution to elliptic equations in divergence form is continuously differentiable provided that the modulus of continuity of coefficients in the $L^1$-mean sense satisfies the Dini condition. This in particular answers a question recently raised by Yanyan Li and allows us to improve a result of Brezis. We also prove a weak type-$(1,1)$ estimate under a stronger assumption on the modulus of continuity. The corresponding results for non-divergence form equations are also established.

17 pages; accepted in Communications in Partial Differential Equations; minor changes in text

References in corpus (1)

• On the $C^1$ regularity of solutions to divergence form elliptic systems with Dini-continuous coefficients

Cited by in corpus (23)

• 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
• On $C^{1/2,1}$, $C^{1,2}$, and $C^{0,0}$ estimates for linear parabolic operators
• On conormal and oblique derivative problem for elliptic equations with Dini mean oscillation coefficients
• Green's function for nondivergence elliptic operators in two dimensions
• Harnack inequality for parabolic equations in double-divergence form with singular lower order coefficients
• Schauder type estimates for degenerate or singular elliptic equations with DMO coefficients
• Characterizations of diffusion matrices in homogenization of elliptic equations in nondivergence-form
• The Dirichlet problem for second-order elliptic equations in non-divergence form with continuous coefficients
• Gradient estimates for singular $p$-Laplace type equations with measure data
• Higher order boundary Harnack principles in Dini type domains
• Note on Green's functions of non-divergence elliptic operators with continuous coefficients
• Gradient estimates for Stokes systems with Dini mean oscillation coefficients
• Hessian estimates for non-divergence form elliptic equations arising from composite materials
• Gradient estimates for divergence form elliptic systems arising from composite material
• Sparse gradient bounds for divergence form elliptic equations
• Regularity of elliptic equations in double divergence form and applications to Green's function estimates
• Schauder Type Estimates for a Class of Hypoelliptic Operators
• Gradient estimates for Stokes systems in domains
• Mean oscillation gradient estimates for elliptic systems in divergence form with VMO coefficients
• A Green function characterization of uniformly rectifiable sets of any codimension
• Refined regularity at critical points for linear elliptic equations
• Gradient estimates for divergence form parabolic systems