Mean oscillation gradient estimates for elliptic systems in divergence form with VMO coefficients

We consider gradient estimates for $H^1$ solutions of linear elliptic systems in divergence form $\partial_\alpha(A_{ij}^{\alpha\beta} \partial_\beta u^j) = 0$. It is known that the Dini continuity of coefficient matrix $A = (A_{ij}^{\alpha\beta}) $ is essential for the differentiability of solutions. We prove the following results: (a) If $A$ satisfies a condition slightly weaker than Dini continuity but stronger than belonging to VMO, namely that the $L^2$ mean oscillation $\omega_{A,2}$ of $A$ satisfies \[ X_{A,2} := \limsup_{r\rightarrow 0} r \int_r^2 \frac{\omega_{A,2}(t)}{t^2} \exp\Big(C_* \int_{t}^R \frac{\omega_{A,2}(s)}{s}\,ds\Big)\,dt < \infty, \] where $C_*$ is a positive constant depending only on the dimensions and the ellipticity, then $\nabla u \in BMO$. (b) If $X_{A,2} = 0$, then $\nabla u \in VMO$. (c) If $A \in VMO$ and if $\nabla u \in L^\infty$, then $\nabla u \in VMO$. (d) Finally, examples satisfying $X_{A,2} = 0$ are given showing that it is not possible to prove the boundedness of $\nabla u$ in statement (b), nor the continuity of $\nabla u$ in statement (c).