Solvability of one Neumann-type problem for 3-harmonic equation in a ball

A boundary-value problem for 3-harmonic equation in a unit ball, containing in the boundary conditions the Laplacian levels up to the second order inclusively, and the normal derivative, is considered. This problem is a natural Neumann-type continuation of the Riquier problem for a 3-harmonic equation. The problem is more general than the considered one, but it has been researched before by V.V. Karachick and B. Torebek for a biharmonic equation. By the means of reducing the initial boundary-value problem to a system of three differential equations of the third order in harmonic equations in a unit ball of functions, the necessary and sufficient condition for solvability of the initial Neumann-type boundary-value problem is discovered. This condition is obtained as a vanishing of the integral over the unit sphere from one of the boundary functions of the problem. Besides, the method of theorem proof allows framing the solution of the considered Neumann-type problem in an explicit form. Moreover, it is determined in the article that solution of the initial boundary-value problem is unique up to an arbitrary constant.