Growing Polyharmonic Functions and The Cauchy Problem for Some Domains in Two Dimensional Euclidean Space

In this paper we consider the Cauchy problem for the second-order polyharmonic equation in an unbounded domain $D$, where boundary data are given on a part of boundary. The goal is to reconstruct a function $v(\eta)$ in $D$ based on the given values of $v$, its Laplacian, and their normal derivatives. This is the ill-posed inverse 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. Additionally, Lavrent’ev regularization is applied to construct an approximate solution. We establish a regularized solution of the Cauchy problem for the biharmonic equation in an unbounded domain. These results provide a foundation for stable reconstruction methods for polyharmonic functions in certain classes of unbounded domains.

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