Regularity For Weak Solutions Of Nonlinear Sub-elliptic Systems With A Drift Term In The Heisenberg Group

The regularity theory for weak solutions is an important and difficult problem in the field of modern partial differential systems,which has a long history of research.Recently,the regularity for weak solutions of nonlinear sub-elliptic partial differential system(s)with a drift term composed of non-commutative vector fields has been widely concerned by mathematicians,which is widely used in many fields of physics,finance,and so on.In this paper,we generalize the classical harmonic approximation theory in Euclidean space to the non-commutative Heisenberg group,and use it to consider the regularity for weak solutions of nonlinear sub-elliptic partial differential systems with a drift term under the controllable growth conditions and natural growth conditions,respectively,while the coefficient Aiα satisfies Holder continuity and Dini continuity.The content of this paper is as follows:In the first chapter,we briefly introduce the research background,research status,research questions and research contents,research methods and innovation points.In the second chapter,we elaborate on the basic knowledge of Heisenberg group Hn,some marks,basic inequalities and lemmas,which will be used later.In the third chapter,for sub-elliptic systems with H(?)lder continuous coefficients in the Heisenberg group.We take advantage of the A-harmonic approximation method to obtain the optimal partial Holder regularity for weak solutions with respect to the first-order horizontal gradient under the super-quadratic(2<m<∞)natural growth conditions and super-quadratic controllable growth conditions,respectively.In the fourth chapter,for sub-elliptic equations with Dini continuous coefficients in the Heisenberg group.We use the A-harmonic approximation technique to obtain the optimal partial C1 continuity for weak solutions under the super-quadratic controllable structure conditions and super-quadratic natural structure conditions,respectively.