On the Unambiguous Solvability of a Multidimensional Initial-boundary Value Problem for the Beam Oscillation Equation with Nonlocal Boundary Conditions in Sobolev Classes

In the multidimensional case, the problem with initial and nonlocal boundary conditions for the beam oscillation equation, considering its rotational motion during bending, is studied. A theorem of existence and uniqueness of the posed problem in Sobolev classes is proved. The solution to the considered problem is constructed as a sum of a series based on the system of eigenfunctions of a multidimensional spectral problem, for which its eigenvalues are found as roots of a transcendental equation, and the corresponding system of eigenfunctions is constructed. It is shown that this system of eigenfunctions is complete and forms a Riesz basis in Sobolev spaces. Based on the completeness of the system of eigenfunctions, a theorem of uniqueness of the solution to the posed initial-boundary value problem is obtained.