Variational Formulation of Boundary Problems

In this chapter, we will focus on the notions of Sobolev spaces of order 1 and of order \(m\ge 1\) . Sobolev spaces are functional spaces. More precisely, a Sobolev space is a vector space of functions provided with the norm obtained by the combination of the \(L^2\) norm of the function itself and of its derivatives up to a certain order. Derivatives are understood in a weak sense, within the meaning of distributions in order to make the space complete.