An Inhomogeneous Polyharmonic Dirichlet Problem With L~P Boundary Data In The Upper Half Plane

By using higher order Poisson kernels and higher order Pompeiu operators, this paper mainly investigates an inhomogeneous polyharmonic Dirichlet problem with L~P boundary data in the upper half plane and proves the uniqueness of the integral representation solution under some certain estimate. The thesis is divided into three chapters.The first chapter mainly introduces the elementary theory of higher order Poisson kernels and Pompeiu operators including usual notation, fundamental theorem as well as modified higher order Pompeiu operators.The second chapter, using the theory of higher order Poisson kernels and boundedness of maximal functions, mainly establishes an estimate for the integral representation solution of homogeneous polyharmonic Dirichlet problem with L~P boundary data in the upper half plane and proves the uniqueness of the integral representation solution under the estimate.The third chapter, using the higher order Poisson kernels and modified higher order Pompeiu operators, mainly studies the inhomogeneous polyharmonic Dirichlet problem with L~P boundary data in the upper half plane and obtains the uniqueness of the integral representation solution satisfied some certain estimate. As a consequence, we also obtain the Green functions for polyharmonic operators in the upper half plane.