CLOSURE OF FINITE FUNCTIONS IN ONE WEIGHT SOBOLEV TYPE SPACE

The description of the closure of finite or smooth finite functions in functional spaces are classical tasks of functional space theory. This task is important in smooth functional spaces such as those of Sobolev, Nikolski, Besov and in their various generalizations. Usually, in a weightless space of smooth functions, the set of compactly finite functions, generally speaking, is not dense. But in the weighted space of smooth functions, for example, in the Sobolev weighted space, with strong degeneracy of the weight, many compactly finite functions can be dense. Therefore, an important issue is the problem of characterizing the closure of compactly finite functions in the weight space under consideration. Here we consider a weighted space of Sobolev type of the second order with three weights and it describes the closure of the set of functions with compact supports.

On the Solvability of the Multidimensional Version of the First Darboux Problem for a Model Second-Order Degenerating Hyperbolic Equation

A multidimensional version of the first Darboux problem is considered for a model second order degenerating hyperbolic equation. Using the technique of functional spaces with a negative norm, the correct formulation of this problem in the Sobolev weighted space is proved.