THE CORRECTNESS OF THE DIRICHLET PROBLEM IN A CYLINDRICAL DOMAIN FOR DEGENERATE MULTIDIMENSIONAL ELLIPTIC-PARABOLIC EQUATIONS

Abstract

The correctness of boundary value problems on the plane for elliptic equations by the method of the theory of analytic functions of a complex variable has been well studied. When investigating similar questions, when the number of independent variables is greater than two, problems of a fundamental nature arise. A very attractive and convenient method of singular integral equations loses their validity due to the absence of any full theory of multidimensional singular integral equations. Boundary value problems for second-order elliptic equations in domains with edges have been studied in detail. In the authors papers explicit forms of classical solutions of Dirichlet problems in cylindrical domains for multidimensional elliptic equations are found. In this paper we use the method proposed in the authors works, we show the unique solvability and obtain an explicit form of classical solution of the Dirichlet problem in a cylindrical domain for degenerate multidimensional elliptico-parabolic equations.