Solutions of the cauchy problem for the polyharmonic equation (uniqueness and approximation)

Abstract

One finds conditions which ensure the possibility of weighted mean-square approximation of a vector-function defined on the boundary of an n-dimensional domain

by vector-functions of the form$\left\{ {\frac{{\partial ^s u}}{{\partial vs}}} \right\}_{s = 0}^{2m - 1}$, where u is, the solution of the equation Δ^m _u=0 in

while∂/∂v denotes differentiation along the normal. The weight function is continuous and positive everywhere on

with the point

whose relative neighborhood

is contained in some (n-1)-dimensional plane. The solution of this approximation problem is closely related with a certain uniqueness theorem for the solution of the Cauchy problem for the polyharmonic equation, also proved in the paper.