On the C regularity of solutions to divergence form elliptic systems with Dini-continuous coefficients

We prove C1 regularity of solutions to divergence form elliptic systems with Dini-continuous coefficients. This note addresses a question raised to the author by Haim Brezis, in connection with his solution of a conjecture of Serrin concerning divergence form second order elliptic equations (see [1] and [2]). If the coefficients of the equations (or systems) are Hölder continuous, then H1 solutions are known to have Hölder continuous first derivatives. The question is what minimal regularity assumption of the coefficients would guarantee C1 regularity of all H1 solutions. See [3] for answers to some other related questions of Haim. Consider the elliptic system for vector-valued functions u = (u1, · · · , uN ), ∂α(A αβ ij (x)∂βu ) = 0, in B4, i = 1, 2, · · · , N, where B4 is the ball in Rn of radius 4 centered at the origin. The coefficients {A ij } satisfy, for some positive constants Λ and λ, |A ij (x)| ≤ Λ, x ∈ B4, (1) ∫ B4 A ij (x)∂αη ∂βη j ≥ λ ∫ B4 |∇η|, ∀ η ∈ H 0 (B4,R ), (2) ∗Partially supported by an NSF grant.