Basic properties and geometric comments of Sobolev spaces

This study consists of eight chapters.In the first chapter, in order to discuss the theory of Sobolev spaces we shall startwith some simple basic notions that are necessary for introducing and studying thesespaces.In the second chapter, we introduce the concept of weak derivative that is a maintool to define the Sobolev spaces.In the third chapter, we review definitions and properties of Sobolev spaces,which are indispensable for the functional analysis.In the fourth chapter, we give a Characterization of Sobolev spaces via theFourier transform.In the fifth chapter, some results are easier to prove over the whole of ?n ; toprove them for extend the setting from to ?n , then restrict it back again to .In the sixth chapter, it is given the embedding theorems and Sobolev inequalities whichare important areas to understand the mathematical properties of Sobolev spaces.In the seventh chapter; even better, some of the inclusions of Sobolev spaces intoother spaces are compact (in the functional analysis sense).Finally in the eigth chapter, It is important to understand what is meant byrestricting a Sobolev function to the boundary of the domain.