Optimal partial regularity for sub-elliptic systems with Dini continuous coefficients under the superquadratic natural growth

The paper is concerned with nonlinear sub-elliptic systems with Dini continuous coefficients under super-quadratic (m>2m>2) natural growth conditions in the Heisenberg group. A general criterion for weak solutions to be regular in the neighborhood of a given point is established by a generalization of method of AA-harmonic approximation. 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 horizontal gradient of weak solutions on its regular set.