Sobolev spaces

Abstract

The goal for this paper is to present material from Gilbarg and Trudinger’s Elliptic Partial Differential Equations of Second Order chapter 7 on Sobolev spaces, in a manner easily accessible to a beginning graduate student. The properties of weak derivatives and there relationship to conventional concepts from calculus are the main focus, that is when do weak and strong derivatives coincide. To enable the progression into the primary focus, the process of mollification is presented and is widely used in estimations. Imbedding theorems and compactness results are briefly covered in the final sections. Finally, we add some exercises at the end to illustrate the use of the ideas presented throughout the paper.