| Summary: | We consider gradient estimates for H1 solutions of linear elliptic systems in divergence form ∂α(Aαβij∂βuj)=0. It is known that the Dini continuity of coefficient matrix A=(Aαβij) 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 L2 mean oscillation ωA,2 of A satisfies
$ X_{A,2} := \limsup\limits_{r\rightarrow 0} r {{\int \limits }_{r}^{2}} \frac {\omega _{A,2}(t)}{t^{2}} \exp \left (C_{*} {{\int \limits }_{t}^{R}} \frac {\omega _{A,2}(s)}{s} ds\right ) dt < \infty , $
where C∗ is a positive constant depending only on the dimensions and the ellipticity, then ∇u ∈ BMO.
(b) If XA,2 = 0, then ∇u ∈ V MO.
(c) Finally, examples satisfying XA,2 = 0 are given showing that it is not possible to prove the boundedness of ∇u in statement (b), nor the continuity of ∇u when ∇u∈L∞∩VMO.
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