The representation of solutions in an explicit form for boundary value problems of differential equations is always very important. In this article the Green function for the Dirichlet problem is constructed in an explicit form inside a sphere for the polyharmonic equations. We note that the obtained formulas have an independent sense. In particular, the explicit representation of the solution of the Dirichlet problem for the polyharmonic equation has important consequences in the theory of elasticity.

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In this paper, we study the correctness in the spaces of S.L. Sobolev of new boundary value problems for quasi-hyperbolic differential equations utttt + Au = f (x, t) (A is an elliptic operator acting on spatial variables). For the proposed tasks theorems on the existence and uniqueness of solutions are proved, and examples of non-uniqueness are given.

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