The Green Function of the Dirichlet Problem for the Biharmonic Equation in a Ball

Abstract

An elementary solution of the biharmonic equation is defined. By using the properties of the Gegenbauer polynomials, series expansions of this elementary solution and an associated function with respect to a complete system of homogeneous harmonic polynomials orthogonal on a unit sphere are obtained. Then the Green function of the Dirichlet problem for the biharmonic equation in a unit ball is constructed in the case when the space dimension n is larger than 2. For \(n > 4\), a series expansion of the Green function with respect to a complete system of homogeneous harmonic polynomials orthogonal on a unit sphere is obtained. This expansion is used to calculate the integral, over a unit ball, of a homogeneous harmonic polynomial multiplied by a positive power of the norm of the independent variable with a kernel being the Green function. The Green function is found in the case \(n = 2\).