Neumann boundary condition for a nonlocal biharmonic equation

The solvability conditions for a class of boundary value problems for a nonlocal biharmonic equation in the unit ball with the Neumann conditions on the boundary are studied. The nonlocality of the equation is generated by some orthogonal matrix. The presence and uniqueness of a solution to the proposed Neumann boundary condition is examined, and an integral representation of the solution to the Dirichlet problem in terms of the Green's function for the biharmonic equation in the unit ball is obtained. First, some auxiliary statements are established: the Green's function of the Dirichlet problem for the biharmonic equation in the unit ball is given, the representation of the solution to the Dirichlet problem in terms of this Green's function is written, and the values of the integrals of the functions perturbed by the orthogonal matrix are found. Then a theorem for the solution to the auxiliary Dirichlet problem for a nonlocal biharmonic equation in the unit ball is proved. The solution to this problem is written using the Green's function of the Dirichlet problem for the regular biharmonic equation. An example of solving a simple problem for a nonlocal biharmonic equation is given. Next, we formulate a theorem on necessary and sufficient conditions for the solvability of the Neumann boundary condition for a nonlocal biharmonic equation. The main theorem is proved based on two lemmas, with the help of which it is possible to transform the solvability conditions of the Neumann boundary condition to a simpler form. The solution to the Neumann boundary condition is presented through the solution to the auxiliary Dirichlet problem.