The Regularity And Rigidity Of Some Dirichlet Boundary Value Problem On The Unit Ball

The thesis is to study the rigidity and regularity problem associated to the solution of a degenerate elliptic second order differential equations in the unit ball Bn in Cn.The regularity theory for the uniformly elliptic partial differential equations[2]witn smooth coefficients are well understood(see,for examples,Evans'book[2]and Ginbarg and Trudinger's booK[6]).However,for the degenerate partial equations,there are many interesting problems remain open and are worth to investigate.The Dirichlet boundary value problems for Laplace-Beltrami operators ?g associated to the Bergman metric on the unit ball Bn is an outstanding example.The concrete content of this article contains three parts:In chapter 1,we will give some preliminary knowledge for our research,including introduce the existence?uniqueness and regularity theory of the general elliptic operator and the definition of Laplace-Beltrami operators?Poisson kernel?Green function on the unit ball Bn.In chapter 2,we mainly study the regularity and rigidity of Dirichlet boundary value problem associated to Kohn-Laplacian operator.WhereWe gave a characterization for(?)when Graham type rigidity holds.In particu-lar,we obtain a stronger result(u is not only pluriharmonic but also holomorphic)(see main theorem:Theorem 2.3.1).We also give a regularity theorem 2.3.2 in this thesis.In chapter 3,we give the form of the solution to the nonhomogeneous Dirichlet boundary value problem associated to new operator.And characterize for a simple(?),giving the solution form of homogeneous Dirichlet boundary value problem.