On a Dirichlet problem for a nonlocal polyharmonic equation

The paper studies the solvability conditions for one class of boundary value problems for a nonlocal polyharmonic equation in the unit ball with Dirichlet conditions on the boundary generated by a certain orthogonal matrix. The existence and uniqueness of the solution to the posed Dirichlet problem are investigated and the Green's function is constructed. First, some auxiliary statements are established: the inversability of the Vandermonde matrix of the mth roots of unity is investigated, then the eigenvectors and eigenvalues of the auxiliary matrix generated by the coefficients of the nonlocal operator of the problem are found, and then the inverse matrix to it is obtained. To prove the uniqueness of the solution to the problem, the commutativity of the boundary operators and the nonlocal operator of the problem is established, and it is shown that if a solution to the problem exists, then this solution is a polyharmonic function. Then the conditions for the uniqueness of the solution to the problem under consideration are obtained. Further, on the basis of the auxiliary statements obtained above, conditions for the existence of a solution to the nonlocal problem are found. The solution to this problem is written out through the solution of auxiliary Dirichlet problems for the polyharmonic equation in the unit ball. Finally, using the well-known Green's function of the Dirichlet problem for the polyharmonic equation in the unit ball, the Green's function of the original nonlocal problem is constructed.