ABSTRACT
The assumption of finite energy solutions leads to the definition of the Sobolev space. In this chapter, we continue to discuss the main facts about Sobolev spaces in the multidimensional context. The theory becomes much more technical than in 1D, and any precise study of the subject would turn this monograph into a text on Sobolev spaces. We shall therefore restrict ourselves to reciting the main results on Sobolev spaces and a few examples illustrating the main points. Occasionally we point out important technical details without really explaining them, in hope that we can motivate the reader to undertake a more systematic study of Sobolev spaces in the future. For an excellent exposition on the subject, we refer to Reference 154. In our discussion, we restrict ourselves to a class of curvilinear polygonal domains (for a precise definition, see Reference 42, also see Exercise 8.1), falling into a more general class of Lipschitz domains.
