Nonlinear Subelliptic Equations On Some Vector Fields

The research of equations with nonnegative characteristic form makes up an important subject in the theory of PDEs. In which the second order degenerate equations associated with square sum operators of Hormander vector fields are the most important classes. People are used to calling them subelliptic operators because they have properties similar to that of classical Laplacian. For linear subelliptic equations many works have been done, as for nonlinear case much less is known so far.Considering three kinds of most important and closely related vector fields, we investigate the maximum principles and L^p estimates to some nonlinear subelliptic equations. The paper is organized as follows:In Chapter 1, we review briefly the development of the subelliptic equations, and introduce the content of this dissertation together with some elementary knowledge which is necessary in this paper.In Chapter 2, we investigate the subelliptic equations associated with Hormander vector fields. Firstly, a homogeneous Harnack inequality as well as a maximum principle of nonhomogeneous subelliptic p-Laplace equations is proved by improving the classical Moser iteration method. The optimal estimate of m is also obtained through log |log| method. Secondly, by using the Sobolev inequality on the C-C spaces we prove a Fefferman-Phong type lemma. Based on which we provide nonexistence theorems to three kinds of nonlinear subelliptic parabolic equations through a relatively unified method.In Chapter 3, we research the subelliptic equations on polarizable Carnot group. First, we introduce the concept of polarizable Carnot group and give some new properties of its homogeneous norm. Then we construct a class of non-divergence equations as well as their nontrivial solutions. The failure of corresponding A-B-P type estimate and uniqueness to the Dirichlet problems in space (?)^Q-ε(Ω) follow. Noting that the C-C distance on Heisenberg group satisfying the eikonal equations, we prove a new kind of Hardy type inequalities with redundant terms. Under some excess convex assumptions on the boundary, we get a Hardy type inequality similar to that in Euclidean space. Finally, we investigate the boundary behavior of solutions to subelliptic p-Laplace Dirichlet problem and get the L^p-estimates of solutions as well as their generalized gradients and Hardy potentials.In Chapter 4, we study nonlinear subelliptic equations on generalized Baouendi-Grushin vector fields. Using the quasihomogeneous properties of these vector fields and their induced metric, we get some results on A-B-P type estimate analogous to that of polarizable Carnot groups. As for subelliptic p-Laplacian, we obtain the growth estimates of its solutions as well as their generalized gradients and Hardy potentials by using an Hardy type inequality proved by D'Ambrosio. Our result shows that the solution is finite vanishing at the origin.