Boundary Value Problems for Higher Order Complex Partial Differential Equations

In this thesis, we mainly study some Dirichlet boundary value problems for higher order complex differential equations in the unit disc. The key tool which we use is the decompositions of polyanalytic functions and polyharmonic functions. At first, we establish a decomposition theorem for polyharmonic functions which is a natural extension of the decomposition for biharmonic functions due to Goursat. As a consequence, we find the polyharmonic analogues of Poisson kernel which are called the higher order Poisson kernels and expressed in terms of some vertical sums with nice structure. Next, applying the higher order Poisson kernels, we obtain the unique solution of the Dirichlet problem for polyharmonic functions in the unit disc (simply, PHD problem). By the decompositions and the result for PHD problem, we give the solutions of three kinds of Dirichet problems for higher order homogeneous complex PDEs in the unit disc. In the last, using the higher order Pompeiu operators and the results for homogeneous complex PDEs, we consider the corresponding Dirichlet problems for inhomogeneous complex PDEs and give their solutions.