Integral Representations of a Function in the S. L. Sobolev Space and Their Application to Boundary Problems

First, we prove a theorem on the integral representation of functions of three variables at the middle of a domain in S. L. Sobolev space with a dominant mixed derivative on a three-dimensional parallelepiped. Further, an integral representation of periodic functions of three variables is given at the middle of the domain in the space of S. L. Sobolev with a dominant mixed derivative. A theorem is also given on the integral representation of homogeneous functions of three variables at the middle of a domain in S. L. Sobolev with a dominant mixed derivative. In addition, a theorem is given on the integral representation of odd functions of three variables at the middle of a domain in S. L. Sobolev with a dominant mixed derivative. Next, we present a theorem on the integral representation of even functions of three variables at the middle of a domain in S. L. Sobolev with a dominant mixed derivative. The above theorems are directly applicable to the qualitative theory of differential equations. In this article, in the most general form, an integral representation of functions of several variables at the middle of a domain in S. L. Sobolev with a dominant mixed derivative on a multidimensional parallelepiped. In this article, such an integral representation of functions in Sobolev space is used to study a boundary value problem in the middle of a domain for the Bianchi integro-differential equation, which is a class of dominating mixed differential equations. For the Bianchi integro-differential equation, the boundary value problem in the middle of the domain in the classical form is reduced to a nonclassical boundary value problem. In this setting, no additional conditions such as matching are required. Then the non-classical boundary value problem posed in the middle of the region is reduced to an operator equation. With the method of integral representations of functions for the boundary value problem, an equivalent integral equation is constructed. Using this integral equation, we prove the homeomorphism theorem. By definition, this theorem is demonstrated by the correct solvability of the considered boundary value problem in the middle of the domain.