задача Дирихле

Well-posedness of the Dirichlet Problem for One Class of Degenerate Multi-dimensional Hyperbolic-parabolic Equations

Authors: Aldashev S. A.

It has been shown by Hadamard that one of the fundamental problems of mathematical physics, the analysis of the behavior of oscillating string is an ill-posed problem when the boundary-value conditions are imposed on the entire boudary of the domain. As noted by A. V. Bitsadze and A. M. Nakhushev, the Dirichlet problem is ill-posed not only for the wave equation but for hyperbolic PDEs in general.

Well-posedness of the Dirichlet Problem for a Class of Multidimensional Elliptic-parabolic Equations

Authors: Aldashev S. A.

Correctness of boundary problems in the plane for elliptic equations is well analyzed by analitic function theory of complex variable. There appear principal difficulties in similar problems when the number of independent variables is more than two. An attractive and suitable method of singular integral equations is less strong because of lock of any complete theory of multidimensional singular integral equations.

Green Function of the Dirichlet Boundary Value Problem for Polyharmonic Equation in a Ball Under Polynomial Data

Authors: Karachik V. V.

The classical Dirichlet boundary value problem for the polyharmonic equation in the unit ball is considered. For this problem with polynomial right-hand side and zero boundary data a polynomial solution is constructed. Our approach is based on the Almansi representation of polyharmonic functions and on the previously obtained an explicit representation of the harmonic components, expressed through the given polyharmonic function. In the case of the harmonic equation the known representation of the solution through the Green function is obtained.