Existence And Regularity For The Dirichlet Problem Of A Class Of Fully Nonlinear Complex Elliptic Partial Differential Equations

In this thesis,the author studies the existence and regularity for the Dirich-let boundary value problem of a class of fully nonlinear complex elliptic equations.The class of equations contain complex Monge-Ampere equations,complex k-Hessian equations,and the linear combination equations of elementary symmetric function composed of eigenvalues of complex Hessian matrices of unknown functions.The last one is called the Quotient equations.The major research of this thesis contains two parts.The first part studies the sharp Holder continuity of the weak solution for the Dirichlet problem of com-plex k-Hessian equations in a smoothly bounded strictly k-pseodoconvex domain of plurisubharmonic type m.The complex k-Hessian equation is the well-known complex Monge-Ampere equation with k=n,it is Laplace equation when k=1,and their weak solutions have many geometric applications.This part of the work is motivat-ed from S.Y.Li's work on complex Monge-Ampere equation,by combining Ngyue's method on k-Hessian equations to obtain a new regularity theorem.This theorem shows that the C?-regularity of the Dirichlet problem of the complex k-Hessian de-pends on the finite type m of the domains ?,which extend the results of S.Y.Li on complex Monge-Ampere equation to k-Hessian equations.In addition,based on the example of S.Y.Li in his paper,this paper gives two new examples of domains ?which are weak(non-strong)k-pseudoconvex(k=2,3)of finite type m,the regularity index a drops significantly when m>2,which shows the gap phenomenon with great interest between the best regularity and the finite type m.The second part studies the existence and regularity of the solution of the Dirich-let problem for the complex fully nonlinear equationin a smoothly bounded domain ? in Cn with ? having smooth sub-solution on ?.The result generalizes Krylov's work and the results of Guan-Zhang on the Rn and spher-ical Sn,respectively.In comparison with the equation form studied by Krylov,there is no sign requirement for the coefficient function ?(z)of ?_k-1 in this paper,which makes the structure of the equation significantly different from Krylov's equation so that we need to look for an admissible solution in ?k-1 cone instead of ?k cone.In order to solve the problem,the original equation needed to be rewritten as a Quotient equation so that it becomes an elliptic equation with concave structure.Under the assumption that ? and boundary data ? having an admissible sub-solution,we may apply the Maximum principle to get global C⁰,C¹ and C² estimates respectively,then we can apply the Evans-Krylov theorem to obtain C²,? estimation.Finally,we can obtain smooth solution in ? by using of elliptic regularity theory,Schauder estimates or the iterative method.From a technical point of view,the main difficulty lies in the boundary C² estimate due to the appearance of the extraordinary coefficients?(z)and ?l(z).When seeking the estimated value,some new difficult terms emerge after taking the derivative of the equation,but the standard skills used to deal with complex Hessian or quotient equations do not work here.Therefore,new estimation techniques and the construction of some new barrier functions are needed to obtain the desired estimation.