In this paper,the fixed_point theorem is used to estimated an asymptotic solution of initial value problems for a class of third nonlinear differential equations which has double initial_layer properties. We obtain th...In this paper,the fixed_point theorem is used to estimated an asymptotic solution of initial value problems for a class of third nonlinear differential equations which has double initial_layer properties. We obtain the uniformly valid asymptotic expansion of any orders including boundary layers.展开更多
Let G be a Carnot group and D={e 1,e 2 } be a bracket generating left invariant distribution on G.In this paper,we obtain two main results.We first prove that there only exist normal minimizers in G if the type of D i...Let G be a Carnot group and D={e 1,e 2 } be a bracket generating left invariant distribution on G.In this paper,we obtain two main results.We first prove that there only exist normal minimizers in G if the type of D is (2,1,...,1) or (2,1,...,1,2).This immediately leads to the fact that there are only normal minimizers in the Goursat manifolds.As one corollary,we also obtain that there are only normal minimizers when dim G 5.We construct a class of Carnot groups such as that of type (2,1,...,1,2,n 0,...,n a) with n 0 1,n i 0,i=1,...,a,in which there exist strictly abnormal extremals.This implies that,for any given manifold of dimension n 6,we can find a class of n-dimensional Carnot groups having strictly abnormal minimizers.We conclude that the dimension n=5 is the border line for the existence and nonexistence of strictly abnormal extremals.Our main technique is based on the equations for the normal and abnormal extremals.展开更多
文摘In this paper,the fixed_point theorem is used to estimated an asymptotic solution of initial value problems for a class of third nonlinear differential equations which has double initial_layer properties. We obtain the uniformly valid asymptotic expansion of any orders including boundary layers.
基金supported by National Natural Science Foundation of China (Grant No.10771102)
文摘Let G be a Carnot group and D={e 1,e 2 } be a bracket generating left invariant distribution on G.In this paper,we obtain two main results.We first prove that there only exist normal minimizers in G if the type of D is (2,1,...,1) or (2,1,...,1,2).This immediately leads to the fact that there are only normal minimizers in the Goursat manifolds.As one corollary,we also obtain that there are only normal minimizers when dim G 5.We construct a class of Carnot groups such as that of type (2,1,...,1,2,n 0,...,n a) with n 0 1,n i 0,i=1,...,a,in which there exist strictly abnormal extremals.This implies that,for any given manifold of dimension n 6,we can find a class of n-dimensional Carnot groups having strictly abnormal minimizers.We conclude that the dimension n=5 is the border line for the existence and nonexistence of strictly abnormal extremals.Our main technique is based on the equations for the normal and abnormal extremals.
