A Study of the Use of the Fractional Laplacian in Extension Problems

Abstract

We obtain the operator square root (􀀀D)1=2 of the Laplacian, called the fractional Laplacian, from the harmonic extension problem to the upper half space. It turns out that this operator maps the Dirichlet boundary condition to the Neumann condition. In this thesis, we extend the work of [2] by establishing the fractional Laplacian using semi-group methods and also providing proofs to certain claims and propositions in [2]. We also study some properties of the fractional Laplacian and relate it to an extension problem.