On The Rigidity And Regularity Theorems For The Solutions Of A Class Of The Degenerate Elliptic Partial Differential Equations

In this dissertation, we study the rigidity and regularity problems for the solu-tions of a class of the degenerate elliptic equations in the domainΩC Cn. LetΩbe a bounded strictly pseudoconvex domain in Cn. Let u=(?)i,j=1n uijdzi (?) dzj be a Kahler metric onΩinduced by a strictly plurisubharmonic function u. Let△u be the Laplace-Beltrami operator with respect to the metric u. Then△u is a degener-ate elliptic operator when u is a complete metric. First we consider the rigidity of harmonic function which is the solution of the following Dirichlet boundary value problem WhenΩis the unit ball Bn in Cn, we characterize the rotational symmetric potential function u of the Kahler metric u=(?)i,j=1n uijdzi (?) dzj so that the Graham type rigidity holds. In other words, If h (?) Cn(Bn) and is also u-harmonic then h must be pluriharmonic in Bn.WhenΩis not the unit ball, the rigidity problem becomes much more difficult. Since the tools for ball case can no longer be applied when domains have less sym-metry. A typical domain on CR geometry is the domain in Cn whose boundary is a real ellipsoid, we obtained some rigidity results there.Secondly, in this thesis, we study the regularity for the solution of the non-homogeneous equation△gu=f,z(?)Bn, where△g is the Laplace-Beltrami operator. We give an asymptotic expansion for the solution of the equation.