ON THE SOLVABILITY OF ONE BOUNDARY VALUE PROBLEM FOR THE POLYHARMONIC EQUATION IN A MULTIDIMENSIONAL BALL

The necessity of studying boundary value problems for elliptic equations is motivated by their numerous practical applications in the theoretical study of processes in hydrodynamics, electrostatics, mechanics, heat conduction, elasticity theory, and quantum physics. For example, the distribution of the potential of an electrostatic field is described by the Poisson equation, while the vibrations of thin plates with small deflections are described by biharmonic equations In this paper, we study the solvability of one boundary value problem for the polyharmonic equation in the unit ball. In this problem, the normal derivatives of the unknown function are prescribed consecutively from the first-order normal derivative up to the (m 1)-th order derivative, while the (m + 1)-th order derivative is additionally prescribed as the last condition. To solve this problem, we reduce the original problem to the Neumann problem by means of a special integro-differential operator. Then, using another integro-differential operator, the Neumann problem is reduced to the Dirichlet problem. We use the fact that the solution of the Dirichlet problem can be represented by means of the Green function and the Almansi representation. As a result of these representations, the necessary and sufficient condition for the solvability of the original problem is obtained.

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