An Introduction to Distributions and Sobolev Spaces, and a Study of the Fractional Partial Differential Operator1+ (-Δ)s⁄2

In this thesis we derive the basic theory for distributions, fractional and classical Sobolevspaces and the direct method of variations, and apply this theory to discuss solutions of the dirichlet problem for a fractional operator related to the laplacian on both bounded open sets and the whole space, for any dimension. We use both the direct methods of variations and Lax-Milgram to find solutions. We also discuss the spectrum of the operator and give a characterisation of all eigenvalues and eigenfunctions onbounded open sets