Nonlinear elliptic systems with Dini continuous coe cientsFrank

0 Introduction This paper will be concerned with the interior regularity theory of weak solutions to nonlinear elliptic systems of second order in divergence form of the following type: div A(x; u; Du) = 0 in : (0-1) Here is a bounded domain in IR n , u taking values in IR N , and the coeecients A(x; ; p) are in Hom(IR n ; IR N). Under natural hypotheses concerning the regularity and the growth of A with respect to p it has been proved by Giaquinta & Modica GM] that weak solutions of (0-1) admit HH older continuous rst derivatives outside a singular set of Lebesgue measure 0 provided (1 + jpj) ?1 A(x; ; p) is HH older continuous in the variables (x;) uniformly with respect to p. In DG] the rst author and Grotowski gave a simpliied (direct) proof of this result avoiding the use of L p-L 2-estimates for the derivative Du of u. The method of proof also gives the optimal result in one step, i.e. if (1 + jpj) ?1 A(x; ; p) is of class C 0;; for some 0 < < 1 in (x;) then u is of class C 1;; outside the singular set. The essential new feature is the use of the harmonic approximation lemma (cf. DS, Lemma 3.3]; see also Lemma 4.1). The aim of this paper is to weaken the assumptions on A with respect to continuity in the variables (x;) and to show a partial regularity result with optimal estimates for the modulus of continuity for the derivative Du under this weaker assumption. To be more precise, we assume for the continuity of A with respect to the variables (x;) that