Correctness of the mixed problem for degenerate multidimensional elliptic-parabolic equations

The first boundary value problem (the Dirichlet problem) for general elliptic-parabolic equations of the second order was first posed by G. Fichera. Further investigation of this problem is given in the monograph by O.A. Oleinik and E.V. Radkevich and works by V.N. Vragov. The correctness (in the sense of unambiguous solvability) of the Dirichlet problem in the cylindrical domain for multidimensional elliptic-parabolic equations is explored in articles by S.A. Aldashev. The first boundary value problem, or the Dirichlet problem, for multidimensional elliptic equations with degeneration at the boundary has been thoroughly studied. However, to our knowledge, the mixed problem for these equations has not yet been investigated. The correctness of boundary value problem solutions on a plane for elliptic equations using the method based on the theory of analytical functions of a complex variable is well-researched. Principal issues arise with similar problems when the number of independent variables is more than two. The attractive and convenient method of singular integral equations loses its power due to the absence of a complete theory of multidimensional singular integral equations. The paper demonstrates the unique solvability and obtains an explicit form of a classical solution to the mixed problem for a class of degenerate multidimensional elliptic-parabolic equations. The proposed method allows for reducing the problem under study to the mixed problem for degenerate multidimensional elliptic equations studied by S.A. Aldashev.