On Solvability of the Neumann Boundary Value Problem for Non-homogeneous Biharmonic Equation

Abstract

In this work we investigate the Neumann boundary value problem in the unit ball for a nonhomogeneous biharmonic equation. It is well known, that even for the Poisson equation this problem does not have a solution for an arbitrary smooth right hand side and boundary functions; it follows from the Green formula, that these given functions should satisfy a condition called the solvability condition. In the present paper these solvability conditions are found in an explicit form for the natural generalization of the Neumann problem for the non-homogeneous biharmonic equation. The method used is new for these type of problems. We first reduce this problem to the Dirichlet problem, then use the Green function of the Dirichlet problem recently found by T. Sh. Kal’menov and D. Suragan.