Solvability conditions of the $\mathcal{N}_2$ Neumann-type problem for polyharmonic equation in a ball

V. V. Karachik

Vestnik Ûžno-Uralʹskogo gosudarstvennogo universiteta. Seriâ, Matematika, mehanika, fizika, Tome 12 (2020) no. 2, pp. 13-20

Résumé

The $\mathcal{N}_k$ Neumann-type class of problems for a polyharmonic equation in the unit ball is considered. The problems of this class generalize both the well-known Dirichlet problem and the Neumann problem. In a number of works, the set of the necessary conditions for the solvability of this problem has been found for the problems of such class, and it has been assumed that the most complete version of the found necessary conditions is also a set of the sufficient conditions for the solvability of the problem. This was a known fact with regard to the $\mathcal{N}_1$ problem. In this study, an assumption that the found set of the necessary conditions coincides with the sufficient conditions of solvability of the $\mathcal{N}_{2}$ problem for a homogeneous $m$-harmonic equation in a unit ball is proved. First, by changing the variables, the $\mathcal{N}_{2}$ problem is reduced to a simpler $\mathcal{N}_0$ Dirichlet problem, the solution to which is considered to be known. Next, the conditions, under which the performed change of the variables is reversible, are found. The conditions found here are connected with the Dirichlet problem's solution having terms of the first order of smallness in the expansion in the neighborhood of zero. Finally, the previously obtained results are used, which concern the value of the $m$-harmonic function in the unit ball in the center of the ball with the values of the normal derivatives of this function at the boundary of the ball. These solvability conditions are transformed to the conditions associated with the values of the integrals over the sphere of polynomials in the normal derivatives of the desired solution on the unit sphere, the coefficients of which are the elements of the arithmetic Neumann triangle. The found conditions coincide with the previously obtained necessary conditions for the solvability of the $\mathcal{N}_2$ problem.

Détail

Keywords: Neumann-type problem, solvability conditions.
Mots-clés : polyharmonic equation

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