\(C^{1}\)-Regularity for subelliptic systems with drift in the Heisenberg group: the superquadratic controllable growth

Abstract

We investigate the interior regularity to nonlinear subelliptic systems in divergence form with drift term for the case of superquadratic controllable structure conditions in the Heisenberg group. On the basis of a generalization of the $\mathcal{A}$ -harmonic approximation technique, $C^{1}$ -regularity is established for horizontal gradients of vector-valued solutions to the subelliptic systems with drift term. Specially, our result is optimal in the sense that in the case of Hölder continuous coefficients we directly attain the optimal Hölder exponent for the horizontal gradients of weak solutions on the regular set.