On the Hayman Uniqueness Problem for Polyharmonic Functions

Abstract

W.K.Hayman [1] proposed recently a problem concerning the inner conditions of uniqueness for functions u(x ₁,x ₂, ..., x _p ) polyharmonic in simply connected regions (G) of the space R ^p. In the present communication the answer to the Hayman problem is given in the case p = 2, G = R ². Our reasoning is essentially based on the connection between polyharmonic functions of two real variables and polyanalytic functions of one complex variable [4]. The existence of a similar relationship between polyharmonic functions of p variables, p > 2, and “polyanalytic” functions defined on Clifford algebras makes, to some extent, plausible the suggestion that a similar approach (with the use of some facts from Clifford Analysis) may turn out to be successful in the case of polyharmonic functions of p variables, p > 2.