An Introduction to Sobolev Spaces

Sobolev spaces were firstly defined by the Russian mathematician S. L. Sobolev (1908-1989) in the 1930s. Several properties of these spaces have been studied by mathematicians till today. Especially existence and uniqueness, asymptotic behavior, blow up, stability and instability of the solution of many differential equations that occur in applied and in engineering sciences are carried out with the help of Sobolev spaces and embedding theorems in these spaces. This book provides a brief introduction to Sobolev spaces at a simple level with illustrated examples. Some of their applications might be relevant both to undergraduate and graduate students, mathematicians, and engineers who have an interest in getting a quick, but carefully presented, mathematically sound basic knowledge in this domain. In this regard the book fills an important gap in that field. There are seven chapters in the book. The first chapter is devoted to basic concepts consisting of metric spaces, normed spaces, inner product spaces, Hilbert spaces and some important results on fixed point theorems. The second chapter is based on Lp spaces. Several important inequalities, embedding property of Lp and some other spaces are presented in this part. In chapter three, we introduce the notion of the weak derivative. There are many numerical example in order the reader to distinguish the weak derivative from the classical one. Sobolev spaces are presented and widely analyzed in chapter four. Chapter five is devoted to Sobolev embedding theorems. The variable exponent Lebesgue and Sobolev spaces are investigated in chapter six. Finally, in chapter seven, we present the importance of Sobolev spaces in light of their application to some differential equations.