Lectures on Sobolev Spaces

In this section we will, very briefly, recall concepts from the theory of dis-tributions that we will need in the sequel. For details, see Kesavan [1], Chapter 1. Throughout these lectures, we will be working with an open set Ω ⊂ RN. Let us briefly motivate our study of distributions and Sobolev spaces. One of the important partial differential equations that we often study is the Laplace’s equation: −∆u = f in Ω together with some appropriate boundary condition. It turns out that in elasticity and structural engineering, the importance of the solution u stems from the fact that it minimizes, amongst ‘admissible functions ’ v, the energy functional J(v) =