Malaysian Journal of Mathematical Sciences, June 2026, Vol. 20, No. 2
Growing Polyharmonic Functions and The Cauchy Problem for Some Domains in Two Dimensional Euclidean Space
Khasanov, A. B., Juraeva, U. Y., Nik Long, N. M. A.
Corresponding Email: umida_9202@mail.ru
Received date: 16 March 2025
Accepted date: 18 November 2025
Abstract:
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 vη 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.
Keywords: biharmonic functions; Carleman function; Phragmen-Lindelof type theorems; Cauchy problem; integral representation.
