A Liouville Type Theorem For The Dirichlet Problems In A Half Space Of The Fractional Laplacian

In this dissertation, we mainly consider the Liouville type theorem of a fractional laplacian in a half space, there are three principle parts in this paper:introduce the Green’s function on the half space and obtained its properties, the equivalence between the differ-ential equation and the integral equation, the Liouville type theorem of integral equation proved by the method of moving plane. We intend to separate the dissertation into four chapters.In chapter one, we introduce the research background of fractional Laplacian and present some basic properties and the motivation. Next, we simply introduce some impor-tant theorems and results which we shall use later.In chapter two, firstly, we recall the Green’s function of differential equations with the Dirichlet boundary conditions on the unit ball. Secondly, we extend the Green’s function to an arbitrary ball and the half space. Thirdly, we obtained its properties which we shall use later.In chapter three, we mainly show the equivalence between the differential equation and the integral equation.In chapter four, first, we use the Kelvin transform, an equivalence of the weighted Hardy-Littlewood-Sobolev inequality and Holder inequality; Next, we use the moving plane method in integral equation to establish a Liouville type theorem for the system.