Well-posedness of the dirichlet problem in a cylindrical domain for degenerating multidimensional elliptic equations

Abstract

For elliptic equations, boundary-value problems on the plane were shown to be well posed in [1] by using methods from the theory of analytic functions of a complex variable. When the number of independent variables is greater than two, difficulties of a fundamental nature arise. The highly attractive and convenient method of singular integral equations can hardly be applied, because the theory of multidimensional singular integral equations is still incomplete [2]. In the present paper, using a method from [3], we establish the unique solvability of the classical solution of the Dirichlet problem in a cylindrical domain for degenerating multidimensional elliptic equations.