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In the paper we study questions about solvability of some boundary value problems with periodic conditions for an inhomogeneous biharmonic equation. The exact conditions for solvability of the problems are found. [ABSTRACT FROM AUTHOR]

In this paper, we consider a model case of stationary vibrations of a thin flat plate, one side of which is embedded, the opposite side is free, and the sides are freely leaned. In mathematical modeling, there is a local boundary value problem for the biharmonic equation in a rectangular domain. Boundary conditions are given on all boundary of the domain. We show that the considered problem is self-adjoint. Herewith, the problem is ill-posed. We show that the stability of solution to the problem is disturbed. Necessary and sufficient conditions of existence of the problem solution are found. Spaces of the ill-posedness of the considered problem are constructed. [ABSTRACT FROM AUTHOR]