Sobolev spaces : with applications to elliptic partial differential equations / Vladimir Maz'ya

The author considers various aspects of the theory of Sobolev spaces, in particular the embedding theorems. At first Maz'ya gives prerequisites and miscellaneous topics related to the theory of Sobolev spaces; some of this material is of independent interest and some is used in the sequel. He deals with various extension and approximation theorems and with maximal algebras in Sobolev spaces. He also devotes a part to inequalities for functions vanishing on the boundary, along with their derivatives up to some order. Then, he studies necessary and sufficient conditions for the validity of integral inequalities for the gradients of functions that vanish at the boundary. Of particular importance for applications are multidimensional inequalities of Hardy-Sobolev type. Basic results are applied to the spectral theory of the Schrödinger operator.
The author also considers the so-called conductor and capacity inequalities, which are stronger than inequalities of the Sobolev type and are valid for functions defined on quite general topological spaces. ... (MathSciNet)