Properties Of Solutions To Subelliptic And Fractional Subelliptic Equations On The Heisenberg Group

The pioneering work of H(?)rmander in 1960 s showed that the “sum of squares”operators constructed by smooth vector fields have hypoellipticity,which plays a great role in the research of degenerate elliptic partial differential equations.Stein proposed the important idea by using analysis on homogeneous nilpotent Lie groups to the research of partial differential equations.Based on the idea,the study of partial differential operators on homogeneous nilpotent Lie groups has developed rapidly.The Heisenberg group is a special case of homogeneous nilpotent Lie groups,and the sub Laplacian on the group has attracted the attention of many scholars.In this paper,we study the properties of solutions to some equations involving the subelliptic operator and the fractional power of subelliptic operators on the Heisenberg group.This thesis consists of the following three parts:In the first part,we investigate the Dirichlet boundary value problem of singular semilinear subelliptic equation on the Heisenberg group.Given a bounded symmetric convex domain,whose boundary satisfies the Wiener criterion,assuming the nonlinear terms in the equation satisfies some conditions,we decompose the above Dirichlet problem into two boundary value problems which are easily handled,and then obtain the regularity and monotonicity respectively.Furthermore,we prove the monotonicity of the original Dirichlet problem.The second part is to consider two classes of equations related to fractional CR covariant operator on the Heisenberg group.The Hopf lemma and maximum principle are proved by constructing different auxiliary functions and using the extension method on the Heisenberg group.By the CR inversion and the method of moving planes on the Heisenberg group,we establish the Liouville theorem to a class of the semilinear equations.To another class of the semilinear equations,we treat the supercritical case and the subcritical case to get a Liouville theorem of the extension problem by means of Hopf lemma and the generalized CR inversion.The third part is devoted to studying the fractional sub Laplace equations on the Heisenberg group.After proving four maximum principles,a direct method of moving planes on the Heisenberg group is provided.We establish the nonexistence in the subcritical case and the symmetry in the critical case for positive solutions to a fractional sub Laplace equation.To the fractional Schr(?)dinger equation on the Heisenberg group,symmetry and monotonicity of positive solutions are given if the solutions satisfy certain assumptions at infinity.To the fractional p-sub Laplace equation,when the nonlinear term satisfies some conditions,we first get the maximum principles and an estimate for solutions near the hyperplane,and then prove the symmetry and monotonicity of solutions.In addition,a Liouville property to the same equation on a half space is shown by using the maximum principles and the narrow region maximum principle.