Sobolev Spaces

Abstract

Sobolev spaces have been very important in the development of partial differential equations. They are based on the notion of “weak derivative”, which defines partial derivatives for $L^p$-functions which are not differentiable in the classical sense. The weak derivative is based on the simple idea of integration by parts. In this way we transfer the burden of differentiation from a “bad” (nonsmooth) function, to a “good” (smooth) function. In this chapter, we do not conduct an exhaustive study of Sobolev spaces. This can be found in the specialized books included in the bibliography (see the Remarks at the end of the chapter).

Pour atteindre les limites du possible, il faut rêver l’impossible.

René Thom (1923–2002), Fields Medal 1958