I will discuss a recent paper with Seick Kim, in which 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 Haim 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 will also be presented.

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