Partial regularity for sub-elliptic systems with Dini continuous coefficients involving natural growth terms in the Heisenberg group

We consider nonlinear sub-elliptic systems with Dini continuous coefficients in divergence form in the Heisenberg group. Based on a generalization of the method of A-harmonic approximation introduced by Duzaar and Steffen, partial regularity of weak solutions for sub-elliptic systems under natural growth conditions is established. In particular, our result is optimal in the sense that in the case of Hölder continuous coefficients we obtain directly the optimal Hölder exponent for the horizontal gradients of weak solutions on the regular set.