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

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 $2<p<\mathrm{\infty }$, 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 $\mathcal{A}$-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 $T=X_i{X}_{n+i}-X_{n+i}{X}_i$.

KEYWORDS:

• Heisenberg group
• $\mathcal{A}$-harmonic approximation
• partial Hölder regularity
• partial Morrey regularity

AMS SUBJECT CLASSIFICATIONS:

• 35J70
• 35J60
• 35B65

Disclosure statement

No potential conflict of interest was reported by the author(s).

Additional information

Funding

This work was supported by the National Natural Science Foundation of China [grant numbers 11771354 and 12061010] and the Natural Science Foundation of Jiangxi Province [grant number 20202BAB201004].