Remarks on $C^{1,\gamma}$-Regularity of Weak Solutions to Elliptic Systems with BMO Gradients

−Dαa α i (x,Du) = −Dαf α i (x) on Ω ⊂ R , i = 1, . . . , N, α = 1, . . . , n (1.1) belong to C(Ω,R). 1 The system (1.1) has been extensively studied. S. Campanato in [2],[3] proved that (under suitable assumptions) Du ∈ L loc (Ω,R ) with λ 2. From a series of counterexamples starting from famous De Giorgi work (see [6] ) it is well known that Du need not be continuous or even bounded (see [8], [10], [13], [16], [17]) for n > 2. Higher smoothness of coefficients does not improve the smoothness of a solution, as there are examples (see [14]) where the coefficients are real analytic while Du is bounded and discontinuous. On the other hand, it follows immediately from so called direct proof of partial regularity (see [7], [5])that if modulus of continuity of ∂ai ∂p β is small enough then Du is Holder continuous. For this