Partial Regularity Of Weak Solutions For Nonlinear Sub-elliptic Systems In Divergence Form

In this paper,we consider sub-elliptic systems associated with non-commutative vector fields.Regular-ity of weak solutions for nonlinear sub-elliptic systems in divergence form is investigated,respectively in the Heisenberg group and the Carnot group.The details are given as follows.The first chapter briefly introduces the research background and research progress of sub-elliptic sys-tems,the research contents,methods and innovations of this paper.In the second chapter,we study nonlinear sub-elliptic systems with VMO-coefficients in the Heisenberg group.we use the technique of A-harmonic approximation to obtain optimal partial H(?)lder continuity for weak solutions to the sub-elliptic systems under the condition of super-quadratic growth(p>2).In particular,The primary model covered by our analysis is the non-degenerate p-sub-Laplace system involving super-quadratic controllable growth terms.In the third chapter,we investigate the quasilinear sub-elliptic systems with VMO-coefficients in the Heisenberg group.By modifing technique of A-harmonic approximation,we show optimal partial H(?)lder continuity for weak solutions to the sub-elliptic systems under the condition of quadratic growth(p=2).In the fourth chapter,we consider nonlinear sub-elliptic systems with Dini continuous coefficients in Carnot groups.By establishing a Sobolev-Poincar(?) type inequality in Carnot groups and using a generaliza-tion of the technique of A-harmonic approximation with the function V,we obtain C~1-partial regularity for weak solutions to the sub-elliptic systems under the condition of sub-quadratic growth(1<p<2).