Polyharmonic polynomials and mixed boundary value problems in the Heisenberg group \(\mathbb H_n\)

Abstract

We give an existence proof for a polynomial solution of the Poisson equation \(L_0 u=q\) where \(q\) is a polynomial in the one dimensional Heisenberg Group. All the polynomial solutions of the polyharmonic equation \(L_0^m u=0\) in terms of harmonic polynomials are determined. In addition, we also discuss the polyharmonic Neumann and mixed boundary value problems on the Kor\'anyi ball in the Heisenberg group \(\H_n\) by inductive method. Some necessary and sufficient solvability conditions are obtained for the nonhomogeneous polyharmonic Neumann problem.