Regularity For Sub-elliptic Systems Associated With H(?)rmander’s Vector Fields

In 1967,H(?)rmander,in his classic article"Hypoelliptic second order differential equations",proposed the famous finite rank condition for C∞ vector fields,and pointed out that under this condition,sum of squares operators composed of vector fields are sub-elliptic.A typical case is the non-commutative basis vector field over the Heisenberg group Hn Xi=?/?xi+2yi?/?t,Yi=?/?yi-2xi?/?t,i=1,…,n satisfy H(?)rmander conditions,by {Xi,Yi}i=1n constitute the sum of the squares of the operator is a degenerate elliptic partial differential operator.Since then,the regularity theory of weak solutions of sub-elliptic systems composed of non-commutative vector fields has become a new hotspot.In this paper,we consider the regularity of weak solutions of nonlinear sub-elliptic systems with different structural coefficients under the condition of controlled growth in a non-commutative vector fieldH(?)rmander’s vector field,which is more general than Heisenberg group,including Dini continuous coefficients and VMO discontinuous coefficients.The specific content of this paper is as follows:Chapter 1 mainly introduces the research background,research status,research content,methods and innovation points.Chapter 2 introduces the basic knowledge of H(?)rmander’s vector field,basic lemma and basic inequality,which is convenient to use later.Chapter 3 considers the weak solutions of elliptic systems with Dini continuous coefficients.The regularity of C1 of the weak solution is obtained by using the A-harmonic approximation method in H(?)rmander’s vector field under the condition of controlled growth of superquadratic(m>2).Chapter 4 studies elliptic systems with VMO discontinuity coefficients.By applying the A-harmonic approximation technique,the H(?)lder continuity of the optimal part of the weak solution of the systems under the condition of superquadratic(m>2)controllable growth is obtained.