Integration by Parts. Weak Derivatives. Sobolev Spaces

Abstract

Weak derivatives are introduced by taking the rule for integration by parts as a definition. Spaces of functions that are in L ^P together with certain weak derivatives are called Sobolev spaces. Sobolev’s embedding theorem says that such functions are continuous if their weak derivatives satisfy strong enough integrability properties. Rellich’s compactness theorem says that integral bounds on weak derivatives implies convergence of subsequences of the functions itself in L ^P.