INTEGRAL REPRESENTATION FOR POLYHARMONIC FUNCTIONS IN TWO DIMENSIONAL EUCLIDEAN SPACE

In this paper we consider the Cauchy problem for the n-order polyharmonic equation in an unbounded domain D, where boundary data are given on a part of boundary. The goal is to reconstruct a polyharmonic function u(y) in D based on the given values of u, its Laplacians, and their normal derivatives. This is the ill-posed problem, hence we use Carleman-type integral representations and stability estimates to ensure well-posed approximations. We construct the Carleman function and prove key inequalities governing its behavior.