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

TL;DR

This paper establishes regularity and weak type-$(1,1)$ estimates for solutions to elliptic equations, showing that continuity and differentiability depend on the modulus of continuity of coefficients satisfying the Dini condition.

Contribution

It provides new regularity results for elliptic equations with coefficients meeting Dini conditions and extends weak type-$(1,1)$ estimates to broader classes of elliptic operators.

Findings

Weak solutions are continuously differentiable under Dini continuity of coefficients.

Weak type-$(1,1)$ estimates are proven under stronger modulus of continuity assumptions.

Results are extended to both divergence and non-divergence form elliptic equations.

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.