On anisotropic Hardy inequalities and their applications

Certain generalizations of the Hardy inequality are obtained for functions in anisotropic Sobolev spaces on $R^n$ and on certain unbounded domains satisfying a horn condition. On the basis of these inequalities the uniqueness of solution for the Neumann problem in an unbounded domain of 'layer' type is proved and the general form of this solution for a class of quasielliptic equations is established. In addition, a theorem on the absence of negative spectrum is proved for a certain class of such equations, considered in $R^n$.