Partial regularity for a nonlinear discontinuous sub-elliptic system with drift on the Heisenberg group: the superquadratic case

Abstract

ABSTRACT In this paper, we consider a nonlinear discontinuous sub-elliptic system with drift on the Heisenberg group, where the coefficients in system are discontinuous and satisfy the vanishing mean oscillation condition and growth conditions with the growth index 2 < p < ∞ , and the non-homogeneous terms satisfy the controllable growth conditions and the natural growth conditions, respectively. The partial Hölder regularity to weak solutions and the partial Morrey regularity to horizontal gradients of weak solutions are proved by the A -harmonic approximation. One of the difficulties in this paper is how to reasonably define the weak solution and avoid the requirement of integrability of Tu. In fact, we used T = X i X n + i − X n + i X i .